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MATH 480-01 (43128): Topics in History of Mathematics
JB-387, TuTh 6-7:50PM
SYLLABUS Spring 2013
John Sarli
JB-326
(909)537-5374
[email protected]
TuTh 11AM-1PM, or by appointment
Text: V.S. Varadarajan, Algebra in Ancient and Modern Times (AMS Mathematical World Vol. 12, 1998) ISBN 0-8218-0989-X
Prerequisites: MATH 252, MATH 329, MATH 345, MATH 355
This is a mathematics course structured around historical developments that produced our current understanding of algebraic equations. Rather
than being comprehensive, we will focus on a few ideas dating from antiquity that will help explain the interpretation of solutions to algebraic
equations in terms of roots of polynomials. This modern interpretation required the systematic development of numbers from their early
representation as geometric quantities to their formalization, more than two thousand years later, as structured algebraic systems. One of our
objectives is to come to an understanding of why we teach algebra the way we do.
The above text will be used as a guide. We will not cover all of it but you should read as much of it as possible. I will supplement its topics
with notes that will appear on my website www.math.csusb.edu/faculty/sarli/
along with this syllabus.
In order to maintain a seminar approach to this course (active participation through discussion) the grading will be based on just four
components, 25% each:
1) First Written Project;
2) First Exam;
3) Second Written Project;
4) Final exam.
As the course progresses I will make suggestions for suitable project topics, some of which will derive from exercises in the text. The exams,
for which you may use your notes, will be require you to implement some of the basic mathematical techniques that we will develop in class. The
date for the First Exam and the due date of the First Written Project will be announced in class and recorded on the website. Guidelines for
project format will also be posted there.
After assessing your performance on each of the four components, course grades will be assigned as follows:
Some important dates:
April 2, University closed
April 12, Late add period ends
April 19, Last day to drop w/o record
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May 2, First Exam; First Project due
May 31, University closed
June 6, Last day of class (Second Project Due)
Thursday, June 13, Final Exam
Constructible Numbers
Before the invention of the number line, real quantities were represented by geometric constructions, typically with straightedge and compass.
More than an entire book of Euclid's Elements is devoted to geometric arithmetic consistent with his postulates for the development of plane
geometry. For example, given a unit length one can construct a square whose side has this length. Then, by the Pythagorean Theorem, a diagonal
of this figure has the property that the square of its length is . From our perspective we would say that
is a constructible number. It is
natural to ask just what numbers are constructible. This is made precise by stating axioms of constructibility consistent with the proof methods of
Euclidean geometry. The purpose of these axioms is to determine which points in the plane are considered to be constructible. From these
constructible points we can then define constructible numbers. For example, the distance between two constructible points is considered to be a
constructible length (which we would associate with a non-negative real number).
Any two distinct points may be chosen and designated constructible, and the distance between them taken as the unit length.
The intersection of two constructible figures results in constructible points.
The line or segment determined by two constructible points is a constructible figure.
A circle with a constructible point as center and a constructible length as radius is a constructible figure.
The following theorems are easily derived from these four axioms:
The line parallel to a given constructible line and passing through a given constructible point not on the given line is a constructible
figure.
The perpendicular bisector of a constructible line segment is a constructible figure.
The circle determined by three constructible points is a constructible figure. (If the three points are collinear then the resulting "circle"
is clearly a constructible line.)
These three theorems were all Euclid needed to do arithmetic with constructible numbers. By arithmetic we mean the operations of
.
Though Euclid did not have access to the coordinate plane, we can use that setting to describe the arithmetic of constructible numbers succinctly.
In fact, we can take it further by interpreting the ordered pair
as the complex number
. This formalism was not fully developed
before the time of Gauss, but we will see that the arithmetic of constructible complex numbers simplifies many of the historical discoveries we
will study.
To get started, let us take the points
and
. By axiom
Then, by axiom
and
as the two points in axiom
the line we call the real axis is constructible, and by axiom
the number
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is constructible (as is any integer), and by T
. These correspond to the complex (real) numbers
the circle we call the unit circle is also constructible.
the line we call the imaginary axis is constructible. Now
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suppose that
and
are constructible real numbers, that is, they have been constructed on the real axis.
Exercise. Show that
Now suppose
is constructible, as are
and
and
. The numbers
.
and
are clearly constructible, by axiom
. By T
the circle through
,
, and
is constructible. From Euclid (III.35) we have
where
is the length of the segment from
to the other intersection of the circle with the real axis. It follows that the product
constructible number. Similarly, by constructing the circle through the constructible numbers
constructible number. We have shown:
, , and
we see that
is a
is also a
The collection of constructible real numbers is closed under the arithmetic operations.
In other words, we can perform arithmetic within the set of constructible real numbers. A set of numbers with this property is called a field, to
use modern terminology.
Note also that the set of rational numbers is a field consisting entirely of constructible numbers.
Exercise. Use T to show that the complex number
is constructible if and only if the real numbers and are constructible. Deduce that the set of all
constructible complex numbers is a field and develop formulas for the sum, difference, product and quotient of two constructible complex numbers.
Geometers of Euclid's time did not have the concept of complex numbers but they discovered many constructions that had an impact on
modern algebra. Here is one that allows us to find the reciprocal of any complex number although it was originally applied only to constructible
points:
Consider a constructible circle centered at
constructible as is the line through
with radius
perpendicular to line
and let
be a constructible point inside the circle, that is,
. Since the points
and
between any two of these four points is constructible. Also, the tangents to the circle at
hence at a constructible point
In the early nineteenth century mathematicians began referring to
in place of
and
is
and
are constructible (why?), and these two tangents meet on the line
,
.
as the generalization of reflection in a line. Note that the right triangles
with
. The line through
where this perpendicular intersects the circle are constructible, the segment
as the inversion of
,
, and
in the given circle and interpreted this construction
are similar to one another (and analogously
). This produces many relations among these constructible lengths, in particular
Thus, the product of the distances from the center of the circle to two points related by inversion is the square of the radius of the circle. The
suggests there is an analogous construction if the constructible point is given to be outside the
symmetrical relationship between and
circle, that is,
(see suggested project 5, below).
Perhaps because they did not have a theory of number systems as we understand it, the ancients were aware of some of the limitations of
geometric arithmetic. In particular, they devoted a lot of mathematics to the attempt to resolve the following four questions regarding
constructibility using only straightedge and compass:
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Q
Is it possible to trisect an arbitrary angle?
Q
Is it possible to construct any regular polygon?
Q
Is it possible to construct a square with area equal to that of a given circle?
Q
Is it possible to construct a cube with volume twice that of a given cube?
Eventually it was proved that each of these questions has a negative answer, but the proofs required an understanding of roots of polynomials
that was not available to the ancients. Questions of constructibility in general are answered by the following theorem, the result of work by Pierre
Wantzel (1814-1848):
If a complex number is constructible then it is a root of an irreducible polynomial with integer coefficients whose degree is a power of
So Q
has a negative answer because, for example, constructing a cube with volume
is a factor of any polynomial that has
constructible because
would require that
as a root. For Q , consider the unit circle, whose area is
.
be constructible. But
. A square with side
is not
is transcendental, that is, not a root of any polynomial with integer coefficients. (This was proved by Ferdinand
Lindemann (1852-1939) working from ideas that Charles Hermite (1822-1901) used to prove that the natural log base
is transcendental.) As
for Q , it can be shown that the above theorem has the following corollary:
be the number of integers less than or equal to
Let
is a power of
that are relatively prime to
. The regular
-gon is constructible if and only if
.
Thus a regular polygon with a prime number of sides is constructible if and only if that prime is of the form
.
Suggestions for First Project:
1. Find the first five primes of the form
smallest
such that
. Prove that
is prime only if
is a root of the polynomial
2. Find an angle such that
trisected with straightedge and compass.
3. Let
at
. Let
be a central angle of the unit circle (centered at
and
be collinear with
show that
4. Let
. Let
be a power of
; find the
such that
. Use Wantzel's theorem to show that the angle
cannot be
) with initial ray the positive real axis and the terminal ray intersecting the circle
is on the circle,
is on the real axis, and
. Explain why this is not a constructible trisection of
and
is a power of
is not a prime.
be constructible positive real numbers. Show how to construct
has unit length. Draw such a figure and
.
. Carry out the construction of a segment whose length is
.
5. Let the constructible point
be outside the unit circle in the complex plane, that is,
the inversion of in the unit circle. Let
be the reflection of
reciprocals of each other as complex numbers.
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in the real line. Show that
. Find a construction for
is constructible and that
and
are
,
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Euclidean Algorithm
Book VII of Euclid's Elements is a departure from the earlier books in that it begins a three-book discussion of what we would now call
elementary number theory. The discussion deals with properties of positive integers and their arithmetic through application of the theorems on
proportions developed in the earlier books. This treatment exhibits a recognition of the natural numbers as, if not an actual number system, at
least a collection if mathematical objects that have systematic properties independent of their representation as geometrical magnitudes. The most
cited result from Book VII is the process for finding the greatest common divisor of two numbers and (a common divisor, usually chosen to
be positive, that is a multiple of any common divisor), which we now commonly denote
. This has come to be known as the
Euclidean Algorithm, a process of repeated subtraction that we now associate with the division algorithm. Here is how it works using the
example
Thus
Given
because
and
with
, let
is the last non-zero remainder in the process. We can represent this process abstractly as follows.
and
. Then the above process would take the form
As Euclid noticed, the process must always terminate after a finite number of steps, that is,
, for some natural number
.
Euclid made special mention of the case when
; we would now say in this case that and are relatively prime. These books on
number theory had great influence on the beginnings of algebra. For example, the work of Diophantus about seven centuries later dealt in a
rather sophisticated manner with finding whole number solutions to problems involving arithmetic combinations of quantities. (This branch of
algebra is now called Diophantine analysis and includes, for example, Fermat's conjecture that was finally settled in 1994 by Andrew Wiles.) As
an example of a Diophantine problem imagine you have two flasks, one that holds
and another that holds
. Is it possible to use
these flasks to measure exactly
backward:
from a source of water into a bucket? To answer this, Euclid would look at his procedure and work
Reversing the procedure this way reveals a lot about the structure of integers. First, it shows that for any two integers
for some integers
and
. (The greatest common divisor does not depend on the sign of the given integers. In number theory we consider both
to be units, so if
then
unique. The point is that if
integers then clearly
the ideal generated by
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divides
and
is usually considered a greatest common divisor as well.) The integers
then there exist integers
such that
, so any integer combination of
is the principal ideal generated by
and
. On the other hand, if
and
and
are not
are any
is a multiple of their greatest common divisor. (We say that
.) This answers our original question: The two flasks do not suffice to
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measure out exactly
because
) because
flask.
is not a multiple of
: Pour
The iterative procedure for finding
steps, for example,
from
. However, we can measure out exactly
measures of the
(and therefore any multiple of
flask into the bucket and then remove
measures of the
has interesting properties that are studied in computer science. For a given number of
as in the above problem,
would be determined in general as follows:
Thus
are expressed entirely in terms of the quotients
, by iteratively removing the remainders
for
.
Suggestions for First Project:
and
6. Find integers
such that
. For integers
in terms of the quotients in the Euclidean Algorithm assuming
and suppose
7. Let
explanation to show that
for any integer
and
are integers such that
are relatively prime if
and
and
, find
and
; do the same assuming
. Explain how
is related to
are relatively prime. Show that
such that
.
and use this
if and only if
.
Euclidean Algorithm in
The study of proportions together with the theory of natural numbers eventually resulted in the formalization of the concept of fractions as
and its associated field of fractions as . Solutions to Diophantine often force the
numbers. Today we refer to the ring of integers
consideration of fractional roots; for example, the attempt to "find a quantity
could be expressed in modern notation as
Because we now take algebraic notation for granted we would solve for
whose cube multiplied by
results in the original quantity"
by factoring:
Early on, the solution
would typically be ignored, and then the "unique" solution might be phrased as "the desired quantity is exactly
half of one unit quantity". The idea of a complete set of solutions
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would not become meaningful until the acceptance of and negative quantities as numbers in their own right. (Note that we usually teach
negative numbers through the concept of "take away" and then only later formalize the idea of taking away a positive number as the addition of
the negative of that number.) In honor of the Euclidean Algorithm algebraists refer to
with
there are integers
as a Euclidean domain, because for any integers
such that
where either
or
. Notice that the condition
is redundant because we insisted
. The condition is stated because
there are many other Euclidean domains that have a natural ordering that is defined only for the non-zero elements. An important example is
, the polynomials in a single variable with rational coefficients: for any polynomials
polynomials
where either
with
not the zero polynomial, there are
such that
is the zero polynomial or the degree of
is strictly less than the degree of
,
Any constant polynomial, except the zero polynomial, has degree zero. The degree of the zero polynomial is undefined. In
of a polynomial plays the same part as the absolute value of a non-zero member of
Algorithm in
? For example, let
in
As in
. Then
, a polynomial of degree
that is not a constant. (In fact,
are called units; they are the only polynomials that have reciprocals in
. What happens when we implement the Euclidean
and
The last non-zero remainder in this process is the constant polynomial
and
and
are relatively prime. For any polynomials
.
we have
and so any polynomial of the form
has the polynomial
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. Thus
is irreducible over
, we write
and say that
, the degree
as a factor. Working the above example backward we find
.) In
and
have no common divisor
the non-zero constants
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and so any member of
can be written as
, for some polynomials
.
Suggestions for First Project:
8. Find
and write it in the form
where neither
nor
is the zero polynomial.
Pythagorean Triples
The problem of finding triples of integers
such that
has a long history, perhaps because the search for all such triples requires only consideration of the natural numbers. The Babylonians studied
right triangles with side lengths
hundreds of years before Pythagoras. Pythagoras pre-dated Euclid by about 250 years and produced
geometric methods for producing certain triples. Euclid showed how to generate triples from any two natural numbers and interpreted his results
geometrically. Diophantus lived about 500 years after Euclid and showed how to produce all Pythagorean triples. In doing so he introduced the
rudiments of algebra though his notation was primitive and inefficient. (Much of the work Diophantus produced did not survive the so-called
Dark Ages in Europe, and that which did was characterized by a lack of generality that left many of his methods somewhat opaque.) Since the
influence of this rather elementary problem continues within modern algebraic geometry we should review this method of producing Pythagorean
triples, using complex number notation. In other words, we will work within the Euclidean domain
, a ring now referred to as the Gaussian
integers because of the extensive contributions Gauss made to understanding their properties.
Let
Then
and
be nonzero integers (there is no loss of generality in assuming they are both positive) and consider the Gaussian integer
Since for any complex number
In other words,
we have
and
is a Pythagorean triple. (Since
it follows that
could be positive or negative we can always work with
as needed.) To an algebraist, two questions immediately arise:
Are all possible triples produced by this method?
Is there a way to produce only those triples with no common divisor?
The second question would ask us to consider
example, we would now say that
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and
and
to be the same triple provided
are equivalent triples and that
is a nonzero integer. For
is the primitive representative of this
.
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equivalence class since all members are of the form
, then
and
. Assuming
cannot both be even numbers (otherwise
odd. So one must be even and the other odd. We say
is odd, and therefore
From now on assume
triple
would all be even). Similarly,
is even. Note that
and
or
since
that
would need to be odd and therefore divisible by some prime
is a prime. But if
contradicts the condition
Let
are both odd and that
could have a common divisor
. This suggests we impose the additional condition on
because any common divisor of
and
cannot both be
have opposite parity.
primitive? Not necessarily because
have the common divisor
either
is primitive and that it is of the form
, of course, is even. Is the
, whereby
would
. Now the triple will be primitive
. But then
divides one of these then it divides the other (since we are assuming that
and so
divides
divides
) and that
. To summarize:
, with
of opposite parity and
. Then
is a primitive Pythagorean triple.
We have not yet answered the first question, which we can now refine: Are all primitive Pythagorean triples of this form for some
?
Euclid did not fully address this, and there are differing opinions as to whether Diophantus produced a rigorous proof that the answer is
affirmative since his system of notation did not allow for more than one unknown and did not admit non-positive solutions to equations.
(Curiously, Diophantus seemed to have no problem with what we would call rational coefficients, and argued that they must be considered as
numbers.) However, Euclid made the connection with angles in right triangles that we can easily see from this complex number representation:
consider the right triangle with legs
and angle such that the ratio of
Given a Pythagorean triple
opposite to adjacent is
; if one bisects
to create a smaller right triangle then the ratio of leg opposite to
to the adjacent leg
is
.
In other words,
Notice this says that the opposite leg of the smaller triangle is
and so the ratio of
to this leg is
Suggestions for First Project:
9. Assuming only that
odd. What does this imply about
Show that any primitive triple
is a primitive Pythagorean triple, not necessarily of the form
and ?
is of the form
. (This is Exercise 7 on page 15 of [Varadarajan].
Addendum: The Modular Group
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, show that
must be
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Since
the Pythagorean triples are points on the cone
and so they are closely connected to rational points on the algebraic curve represented by this cone. The study of rational points on algebraic
curves is called arithmetic algebraic geometry, which is foundational to modern number theory. An important tool in this subject that appears in
many branches of mathematics and physics is the modular group
entries. Given
, which consists of
matrices of determinant
with integer
the matrix/vector multiplication
extends to matrix/vector multiplication on
by identifying
which preserves the quadratic form
with the matrix
and so leaves the cone invariant. Further, if the triple
then the triple
is
given by
This is the spin representation of
. If
can be shown that any coprime pair can be produced as
, whereby any Pythagorean triple on the cone can be produced as
Note that
and
have opposite parity if
are both odd and
for some
.
are both even; the matrices with this property form a
subgroup of
.
Suggestions for First Project:
10. If
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show that
provided
. (Note that the determinant
condition
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implies
.)
Pell's Equation
A not too infrequent occurrence in mathematics is the attribution of credit to a mathematician for work that was developed by others. For
example, Leonard Euler (1707-1783) named a particular class of Diophantine equations after the English mathematician John Pell (1610-1685),
a contemporary of Fermat. Perhaps because of Euler's status in regard to solving quadratic Diophantine equations, the attribution stuck and is still
used today for equations of the form
where the coefficient
is a non-square positive integer and
is a non-zero integer. In fact, for any given
, the above equation is often
denoted
. Euler may have meant to credit Fermat but in any case, the techniques for solving
were developed by mathematicians who
flourished in India during the European Dark Ages. Brahmagupta (seventh century) knew that equations of this type have infinitely many integer
solutions, and believed that integer solutions exist for any
(Lagrange finally proved this in 1766). Brahmagupta did a great deal of work on
quadratic equations in general, as did Aryabhata a century earlier. The contributions of these two Indian mathematician/astronomers advanced the
theory of numbers by developing precise approximations of square and cube roots.
Perhaps the most significant advance by Brahmagupta was due to the way he viewed solutions to
. He developed methods of generating
new solutions from known solutions that foreshadowed the modern theory of algebraic groups. An algebraic group is a variety (points that solve
a collection of polynomial equations in several variables) that is also a group, that is, points can be multiplied and have inverses. A simple
example of an algebraic group is the unit circle
where we represent the points as complex numbers
The group product is just complex multiplication, so if
Also,
and
then
because
Note that the rational points on the unit circle form a subgroup. For example, if
and
then
which allows us to interpret signed Pythagorean triples as a group by looking at their projections onto the unit circle. Brahmagupta did not
as curves. But these ideas occur to us naturally
have complex numbers to work with nor did he likely think of quadratic equations such as
and we will see that his methods previewed ideas from algebraic geometry.
The method that Brahmagupta perfected is based on the observation that two known solutions can be composed to create another solution, a
process he called bhavana (the Sanskrit word for "production"). His motivation was probably an attempt to approximate square roots (see page 18
in{Varadarajan]). For a given , consider a solution
to
and a solution
to
Defining
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we have
and we note that
Thus,
is a solution to
, which we might symbolize as
We can symbolize this result by saying that
and
imply
. In particular, if
then
is another solution to
, and if
we see that the set of solutions to
is closed under composition (compare this
with the unit circle group, above). These observations represented a significant advance in algebraic thinking. We would naturally interpret this
result in terms of analytic geometry; for example, if
Since
solves
it follows that
is also solution. But then
together because
In modern language we would say that the algebraic curve
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we could graph the hyperbola
is another solution, etc. The equations
is a group and
and
is a subgroup of index
were often considered
, its other coset being
.
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and
Brahmagupta's work prompted Fermat, Lagrange and many others to ask if all solutions could be generated from a small number of them.
These questions anticipated the development of group theory that began in earnest after the work of Galois and Abel. Here the work of many
mathematicians came together to advance modern algebra. Notice that solutions can be considered as members of
which is an integral domain under ordinary multiplication:
This may have been how Brahmagupta arrived at the bhavana, but in any case we have a close connection with the Gaussian integers
which we would obtain if we allowed
. (The related fields
, where
can be any integer, provide a rich collection of
examples for understanding the the theory of polynomial roots that Galois created in 1830. For which
the integral domain
actually a Euclidean domain is a difficult one and still the focus of much research in number theory.) With
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if
is
, notice that the equation
becomes
which is a circle of radius
,
, and that the bhavana becomes the rule for multiplying complex numbers.
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What we now call representation theory began when algebraists such as Cayley and Hamilton experimented with using matrices to do
arithmetic within integral domains. For example, given complex numbers
imaginary numbers:
Similarly, in
the product
Note that if
and
solve
and
there product can be represented without using
can be represented by
then these matrices have determinant
.
Suggestions for Second Project:
we have shown that the collection of points with integer coordinates on the curve
11. For any given
composition. Show carefully that these points form a group, as follows:
a) Show that the composition is associative, that is,
b) Find the identity element, that is, the point
for all
such that
.
c) For any solution
where
is closed under
, find its inverse, that is the point
such that
is the identity element.
By substituting arbitrary values
and
we obtain a solution to
where
. This is the idea behind the solution technique called the chakravala that Brahmagupta and others developed for
solving the fundamental equation
Starting with initial positive integers
values of
the chakravala generates values
. It was noticed that this process eventually generates
inductively that produce solutions to
and thus solves the fundamental equation
process terminates after a finite number of steps was not proved. Rather, it was justified by noticing that the
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for determined
. That this inductive
could be used to provide
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increasingly precise approximations to
The
and
. This was made precise by Lagrange who proved in 1766 that
are strictly increasing sequences of positive integers that are the convergents to the continued fraction expansion of
The chakravala always terminates after a finite number of steps in a solution
This solution
to
generates all positive integer solutions to
.
under cyclic composition (with itself).
Over the centuries there were many ways to implement the chakravala, but the process can be summarized as a straightforward inductive
algorithm using modern notation:
1) Let
2) For any
and choose
, determine
. The we have
with
.
such that
and let
Thus, it can be shown that
1) We have
2) Since
at some stage of the process. As an example, we use this algorithm to solve
since
,
, and
gives us
.
or
, but
narrows it down to
3)
and
give us
, so
4)
and
give us
, so
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, and so
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Checking, we have
Note also that
which are increasingly better approximations to
.
These are the first four convergents to the continued fraction for
,
for
, which is the standard notation for the sequence of fractions indicated above. Lagrange showed that the convergents to the continued
fraction for
contain all positive integral solutions of
; for example,
shows up as the convergent
.
one notices that
The inductive process behind the chakravala probably was motivated by the bhavana composition. Starting from
which solves
with
so a reasonable next step would be to choose
continues based on similar considerations.
as close as possible to
such that
is divisible by
. The process then
Suggestions for Second Project:
12. Find the fundamental solution
convergent that is equal to
to
. Determine the simple continued fraction expansion of
.
The inductive process behind the chakravala probably was motivated by the bhavana composition. Starting from
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and find the
one notices that
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which solves
with
so a reasonable next step would be to choose
continues based on similar considerations.
If
such that
is divisible by
. The process then
is divisible by
so then
To choose
as close as possible to
is also divisible by
as close as possible to
and
.
First Exam Topics
This is a test where you should use your own notes and be able to:
Explain whether a point in the plane is constructible or not.
Apply the Euclidean Algorithm in
divisor.
and in
and find corresponding combinations of two elements that produce the greatest common
Find the complex "square root" of given Pythagorean triple.
Apply the bhavana to integral solutions of Pell's equations and exhibit the combination as a solution to such an equation.
Implement the chakravala algorithm to find the fundamental solution to
for a given
.
First Project Guidelines
Select one of the suggested problems or discuss one of your own with me.
Start with a brief introductory paragraph explaining the problem and how it fits into the course.
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Explain your solution clearly, in full sentences.
Cite any references you use. If from an internet source, cite the actual published references.
Print out your final draft or send it to me as a PDF. If you do not have math software then write it up clearly in a "bluebook" using only one
side of each page.
Early Algebra of Polynomial Equations
During the European Dark Ages, Hindu-Arabic mathematics continued to flourish. In particular, our modern system of decimal notation was
essentially established about a thousand years ago, after a long development that dates back to the earliest Chinese notation and systems of
writing in India at the time of Euclid. The work of Al-Khwarizmi in the ninth century was particularly influential in the establishment of Algebra
as a discipline distinct from Geometry. Though he did not use modern algebraic notation it is clear that his treatment of algebraic equations in
one variable emphasized solution methods that apply to all equations similar to a given one. Notation we would consider modern was not
established until the work of Francois Viète in the sixteenth century. He was trained as a lawyer and performed duties for the government
comparable to the English code decipherers in WWII, using algebraic methods few could comprehend. He wrote equations using Latin words that
translate in a fairly straightforward manner into modern symbols. For example, he wrote
for
which he would solve as follows:
or
as he would write it, the first occurrence of what we would call the quadratic formula.
Even though the Hindus introduced negative numbers early on, prior to the sixteenth century the reluctance to using them as "coefficients"
resulted in grouping equations by type into categories we would now consider equivalent. (Even Viète grouped quadratics by whether or not the
radicands turned out positive, negative, or zero, which we often still do when teaching them for the first time.) For example, Al-Khwarizmi used
verbal categories to distinguish the cases
His contemporaries did the same and many of them insisted on geometric foundations to algebraic solutions that should be based on Euclid.
For example, Thabit ibn Qurra justified the solution of
by drawing a square with vertices
and saying that
represents ,
shows that
where
represents
is the midpoint of
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, and
represents
, where
is the vertex of a rectangle obtained by extending segment
. He then
. He also provided geometric "justifications" for the other two quadratic types given by Al-Khwarizmi.
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Another contemporary, Abu Kamil, worked freely with not only negative coefficients but also non-rational square roots. He would show these
could similarly be given geometric interpretations but took the view that these were not required for justification. This viewpoint, working freely
with the arithmetic of coefficients and solutions, encouraged mathematicians at the time to discover many identities that would make possible the
work of later algebraists, such as these two due to Al-Karaji:
By the twelfth century, trade between the West and the East rekindled interest in Mathematics in Europe. Mathematicians became reacquainted
with methods for solving quadratics and associating various interpretations with these solutions. The possibility of finding constructible
quantities from given quantities started to be analyzed in terms of roots of polynomials. For example, the possibility of trisecting a given angle
might be associated with the solutions of a cubic equation. The identities of Al-Karaji and others were realized as solutions of polynomial
equations of higher degree, at first by working backward; for example, noticing that
solves
but that it apparently it also solves
raised questions about the relation between these two polynomials that would lead to the formal concept of factoring and the extension of the
Euclidean algorithm to polynomials. The consideration of non-real roots would occur much later, but Fibonacci (Leonardo of Pisa) made use of
such identities to describe quantities that were not necessarily constructible, yet they arose from simple inductive processes. The most
remembered of these is the sequence attributed to him, the historical importance of which survives in the form of two modern concepts:
1) Recursion relations;
2) Exponential growth.
The Fibonacci sequence
is generated, to use modern terminology, by the recursion relation
which suggests the generalization
It is typically shown in a basic combinatorics course that any such sequence can be "solved" by expressing
where
in the form
are the roots of some quadratic polynomial. Clearly such relations make sense over the entire field of complex numbers (as well
as other domains). For example, with
we obtain
which would seem to fit no obvious pattern until we introduce a technique often used in solving differential equations: Assume
some non-zero
to be determined. Then
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for
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There will always be two complex values
(possibly equal) that satisfy this quadratic relation. But then, for any constants
because
So
and so, if
satisfies the relation
for
. From the given conditions we have
,
For the original Fibonacci sequence,
so
so
Whereas, for our sequence with the complex numbers,
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, so
we have
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Thus
for example,
Notice that
can have non-real roots for real values of and , so the power representation of
can require complex
numbers even if all members of the sequence are real. Fibonacci did not contemplate complex representations, but he did notice in general that
sequences of this type grow at a rate we would describe as exponential when considered as a function of the index.
Suggestions for Second Project:
13. a) Carry out in detail Thabit ibn Qurra's geometric interpretation of the solution to
b) Assuming
and
are positive real numbers and that
.
denotes the real cube root of any real number
, prove Al-Karaji's identity
and find a polynomial for which this quantity is a root.
14. Find a recursion relation with real coefficients for which
for this sequence. Find a relation for which
15. The polynomial
to find the general term
and
. How do you express
, where
in terms of
are non-real complex conjugates and express
in the form
in terms of powers in this case?
are the roots. Solve the recursion relation
and .
The Cubic Equation
Solutions to cubic equations of the form
and their variants were considered by mathematicians of Euclid's time and even extended to equations of higher degree by the time of
Diophantus. Though non-integer solutions were taken seriously by Hindu and Arabic mathematicians a thousand years ago, the reluctance to
admit coefficients that were not positive numbers persisted through Fibonacci's era. Fibonacci himself worked on cubic equations, but it was not
until work of Italian algebraists in the fifteenth century that the concept of polynomial algebra was accepted, and with that acceptance the
development of general, abstract methods took hold. The most significant contributions were made by the following names:
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Islamic algebra was first translated into Latin in the twelfth century. Along with this process came the intent to convert to the Hindu-Arabic
system of numeration, including algorithms for computing with decimal notation. The school that promoted this conversion in Europe became
known as the Italian abacists. Pacioli was among its leaders. He himself did not produce much original mathematics, but he was a rigorous
archivist and was responsible for scoping out what was known and what was important to yet discover. In particular, he wrote in 1494 that he
believed a general procedure for solving cubic equations was obtainable along the lines of the quadratic formula. He challenged the mathematical
community with this task, despite the protestations of many algebraists that finding such a solution was as unlikely as carrying out the
"impossible" geometric constructions of the ancients. Within the next twenty years, del Ferro produced an algebraic solution to equations of the
, but he kept this knowledge among his pupils and immediate colleagues. Nonetheless, as word began to circulate Tartaglia
form
claimed knowledge of the general solution to equations of the form
which prompted one of del Ferro's students, Antonio Maria Fiore, to challenge Tartaglia to a contest. He posed several practical problems that
, which Tartaglia generally was able to solve. For example, the equation
reduced to solutions of cubics of the form
resulted from one of these problems. Today we would note that there is exactly one "real" solution
but at the time the "nature" of solutions to such equations was an idea yet to be made precise. In fact, the general absence of either an
term or an term in these equations indicates some awareness of conditions that would result in a positive real soultion but which might be
difficult to characterize in all cases. It should be remembered that the concept of factoring polynomials as freely as we factor integers was not
well developed at this time. Rather, different "types" of equations were associated with solutions of one character or another. For example,
was seen to have
that
as a solution, but, perhaps extraneously,
and
, were also observed as "solutions" by most. Note
On the other hand,
is solved by
, though it was noted by this time that solutions involving real radicals could be produced, namely
:
Finally, "sophisticated" examples such as
were noted. Here,
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solves the equation and the other two "solutions" involve roots of negative numbers:
:
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The legacy of mathematicians such as Al-Karaji gave the Italian abacists confidence in working with such expressions, but now the task was to
make sense of them as numbers and to develop consistent algorithms for calculating with them. Bombelli made the breakthrough that resulted in
a systematic description of complex numbers. He performed many computations with complex quantities by trusting in the power of algebraic
rules; for example, he showed how to divide a real number by the complex number
using the technique of multiplying both numbers by
. He was thus able to discuss what came to be called "Cardano's Formula" for the cubic in great generality, introducing a designation for
complex quantities equivalent to the modern notation, though wordier.
Suggestions for Second Project:
16. Any cubic polynomial with real coefficients has at least one real root. Assume
has exactly one real solution
a) Show that
where
is a quadratic polynomial in
and that
. Suggestion: Start by writing
.
can have either three real solutions or exactly one real solution. If
b)
are positive real numbers.
is the unique real solution show that
and that
. If all three solutions are real show that one is negative and two are positive.
Del Ferro's Formula
For positive numbers
and
, the cubic equation
was known to have a single real solution, clearly positive. If there were no linear term the solution would simply be
. Possibly due to familiarity with Fibonacci's work on sequences as well as the identities Al-Karaji's such as
del Ferro assumed a solution of the form
for some
that could be expressed in terms of
and
Thus, if this form really is a solution it must be the case that
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. He then worked backward, starting by cubing this form:
, the real cube root of
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Since
note that
, del Ferro must have realized that
and
are required to have opposite signs for this to work. To solve for
and
,
Let
Then we can take
which yields
Probably because he preferred to work with positive radicands, del Ferro presented his formula in the equivalent form
Returning to the "challenge" problem
, we have
so
From this we see that Tartaglia's solution anticipated del Ferro's formula, whether he discovered it on his own or not.
The quantity
is called the discriminant of the cubic. I was eventually seen that the sign of
the solutions in a manner analogous to the discriminant of a quadratic equation. Clearly, if
we have
determines the nature of
are both positive, as we have assumed thus far,
and the equation has a unique real solution. After discussing Cardano's formula we will see that a positive discriminant always
implies a unique real solution, regardless of the signs of
and
, and that a negative discriminant always implies three real solutions. What if
? Consider the example
for which we easily see that
and so
and
are solutions. In fact,
is a "double" solution. From the graph
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we see that the curve is tangent to the
-axis at
. A vanishing discriminant will always indicate a "multiple" root.
Suggestions for Second Project:
17. Let
be a constant.
a) Expand
polynomial in
and make a substitution of the form
that removes the square term. Compute the discriminant
for the
.
b) Choose a real value for
. Let
and compute
and
. What happens on the graph when
?
Cardano's Formula
Suppose
is a solution to the cubic equation
In order to obtain a formula without restrictions on the signs of the real numbers
derivation of del Ferro's solution:
This form, usually called Cardano's Formula, has the advantage of expressing
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and
, return to the original form we obtained in the
symmetrically as a sum of terms
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Note that
precisely when
which is only an issue when
For example,
, in which case a non-negative discriminant means
for the equation
Cardano's formula produces
The radicands of both terms are positive numbers and we can extract the real cube root of each to obtain a positive number that does in fact
and combining terms using the rules of real radicals.)
solve the equation. (Check this by expanding
To understand the actual advances Cardano introduced we should summarize briefly what was understood at the time about cubic equations,
using modern notation for clarity.
1) Any cubic equation
can be reduced to the form
by the substitution
.
Thus, adding
to any solution of the reduced equation produces a solution of the original equation. However, this was generally only
applied to real solutions since computations with square roots of negative numbers were not yet systematized.
2) The discriminant
was understood to determine the nature of the solutions, in particular, the case
unique real root, a fact independent of whether
implied a
is positive or negative.
3) If
it was generally believed that the equation had three real solutions, but the interpretation of
was not yet formalized to the
point where all three could be computed systematically.
4) The case
was not yet interpreted in terms of multiplicity of solutions, though it was used to compute a real solution from either del
Ferro's or Cardano's expression for
.
In the Ars Magna, Cardano made significant advances in the interpretation of solutions that would eventually lead to the development of the
algebra of polynomials as we understand it. With regard to 1), he realized that considering the general cubic equation
was important to understanding solutions other than the one produced by his formula, and that the substitution used
to remove the
why
term was just one of many that could be used to analyze solutions in general. For 2) and 3), he gave convincing arguments
produces a unique real solution and
produces three real solutions, which to him implied the need for a larger number
system within which one could consistently interpret
so as to distinguish the cases. (Before Cardano's death Bombelli would develop
systematic computations with what would come to be known as complex numbers.) He was one of the first to realize that cases where three real
roots could be found by inspection indicated that all three must be deducible from his formula; for example
is solved by
,
, and
which must then have solutions
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. Here,
,
, and removing the
, and
term yields
, as is easily verified. However, the formula gives
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an "irreducible case" in his terminology which yet must somehow represent all three real solutions.
As for 4), Cardano offered interpretations in order to explain how the discriminant could change from negative to positive by making slight
changes in the coefficients. This approach influenced the development of the function concept, leading eventually to the continuum view of the
real numbers and the important technique of studying small "perturbations" of solutions that became essential to the development of calculus.
Analysis of the Discriminant
Before explaining all solutions to the cubic in terms of Cardano's formula it is worth examining the details of how
those solutions when
and
determines the nature of
are real. To summarize:
unique real solution
three real solutions
solutions with multiplicity
We will assume knowledge of factoring and basic calculus so that the analysis is not unnecessarily complicated. Let
and assume
If
. Then
it follows that
is a solution of
However,
and so
because
divides
implies
We would say that
and
, and
factors as
is a root of multiplicity 2, a double root, of the polynomial
is a third root. Since
, whereby
. In fact,
but
we cannot have
is a root of the polynomial
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. This analysis has shown that
, so this third root differs from the double root. Conversely, if
, of multiplicity 3. Look again at the example
then
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which Cardano probably tried to interpret using his formula
Such considerations led him to surmise that perhaps a larger number system would permit various interpretations of
and consider again
Next suppose
If
then
for any real value of
the rest of the analysis, then, we can assume
Since
.
and so
. Now
for precisely one real value of
. Of course, in this case
. For
has two distinct critical points, at the real values
, the product
Thus, if
then
then
and
and
have the same sign, and so there is precisely one value of
have opposite signs, and so there are three distinct values of
such that
such that
; whereas if
.
Complex Interpretation of Cardano's Formula
By considering the general form of the cubic equation
Cardano discovered many relations between the coefficients and the solutions that would, two centuries later, lead to Lagrange's theory of
he was able to make these relations explicit. In hindsight, we can use factoring to understand the
resolvents. In particular, for the case
most basic of these relations. Suppose
,
,
are the three real solutions. Then
In this form the coefficients are seen to be symmetric polynomials in the roots of the polynomial that defines the equation. Further, the degree
of the symmetric polynomial plus the degree of the term is always equal to . The Fundamental Theorem of Algebra, first proved by Gauss
(1777-1855), allows us to generalize this observation to polynomials of degree
The FTA asserts the existence of complex numbers
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by factoring them into linear polynomials:
that allow this factorization. This justifies Cardano's speculation about cubics and
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supports Bombelli's development of a system of numbers with possibly non-real parts, but goes further in that the coefficients
themselves be complex numbers, whereby
Here,
is the elementary symmetric polynomial of degree
. For example, if
can now
then
and
is the coefficient of the linear term. Note that
symmetric in the roots of the polynomial:
. The FTA also provides a general definition for the discriminant that is
This definition applies to any complex polynomial, but note what happens when we focus on cubics with real coefficients. For example,
precisely when two of the roots are equal, as expected. However, if all three roots are real and distinct then clearly
. Whereas, if
there is exactly one real root then the other two are complex conjugates of each other. In this case let
let
, the complex conjugate of
Let
with
be the real root, let
and
. Then
real. Then
This is just the reverse of the discriminant condition we have been using but it is simpler to define. Thus Lagrange and others introduced this
definition, which for the polynomial
becomes
. Note that if the coefficients of the original
polynomial are not real then
need not be real. For example,
is solved by
with
component in the eventual development of Galois Theory.
. The interpretation of
in the most general case was a key
We still need to apply the theory of complex numbers to a consistent interpretation of Cardano's Formula. Recall that we formally expressed
solutions to the equation
as
whereby
However,
we conclude that
. Expanding and recombining we obtain
, and since we can assume we are not in the case where
is a solution
This was del Ferro's original observation, but now we can interpret this condition in terms of complex roots. Look again at the example
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for which Cardano's Formula yields
By the FTA,
and
each have three distinct values, so formally there are nine possible sums. But the condition
we choose the values so that
requires that
This restriction eliminates all but three of the sums. Specifically, write
Then
where
is the cube root of
in the first quadrant,
, and again
. Now
number
Similarly,
denotes the complex conjugate of any
. Thus if we choose
implies
and
? Since in each case we have
because
implies
because
then we must choose
.
. Which pairs produce the corresponding roots
we find
( was chosen in the first quadrant) and
is the only positive root. Now
and
. To decide which is which, consider the polar form
so
is close to
because
to
is close to
. Similarly, the argument of
. We conclude
Suggestions for Second Project:
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. Thus
, the angular argument of , is close to
is close to
. Since
and so the argument of
it follows that
is close
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18. Find all solutions to
in two ways:
a) Factor
and find the roots. Compute the Lagrange discriminant
and use it to obtain the Cardano discriminant
.
b) Remove the
term by substitution. Then solve the cubic using Cardano's Formula. Show that the same solutions are obtained as in a) by
pairing the cube roots correctly.
Quartic Equations
Lodovico Ferrari (1522-1565) was a student of Cardano. He found a general solution to fourth-degree polynomial equations, now called
quartics but known then as biquadratics as a result of the method Ferrari introduced. Cardano included Ferrari's solution in the Ars Magna,
listing twenty types of quartic equations. For the general polynomial
the
term can be removed as usual by the substitution
, so Ferrari worked with equations of the form
Having noted that such equations with no linear term could be treated by completing the square, Ferrari applied this strategy even when the
linear term was present. To accomplish this he needed to introduce an auxiliary variable and use the identity
The right side can now be substituted into the original equation to obtain
Since the left side is a perfect square, the idea is to choose the auxiliary
occur if the discriminant of the quadratic is zero:
Thus, we can choose
so that the right side is also a perfect square quadratic. This will
be any value that solves
Sometimes at least one real solution to this auxiliary cubic is evident without first removing the
can be found using Cardano's Formula. Since
term. But in any case a particular solution
produces a perfect square quadratic, we have
Each of these quadratic equations has two solutions and so we obtain four solutions, counting multiplicity if necessary, to the original quartic.
As an example, consider
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which has the auxiliary cubic
Cardano's Formula produces
resulting in three real values for
conjugate. Taking this one for
, one of which is the positive value obtained by adding the cube root in the first quadrant to its complex
(because
is then real) we have
and then, using the quadratic formula,
Of course, the exact value of
is cumbersome to insert into these expressions for
for this choice of cube root is approximately
is not difficult to see that the real part of
four values of
that solve the quartic are approximately
approximations will produce
when substituted into
and
, and therefore
. However, it
. Thus the
, all four of which
. In any case, it is easy to see using calculus that the function
has no real zeros.
Suggestions for Second Project:
19. The equation
can be solved using DeMoivre's Theorem by noting that
and then solve the equation by removing the
rational
. Find the solutions as roots of unity
term and using Ferrari's method. (Note: The auxiliary cubic factors in this case, providing a
without using Cardano's Formula.)
Lagrange Resolvents
Lagrange (1736-1813) understood the importance of working with all roots of a polynomial, which in general will be complex numbers, in
order to understand their relation to the coefficients. If
are the roots of a quartic his discriminant takes the form
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From Ferrari's solution we can take the roots of
to be
and then we find
Since
is a root of the auxiliary cubic we can use the relation
to rewrite
in terms of the coefficients
only, by continually replacing powers of
we reach the following multiple of the denominator:
For the above example we had
auxiliary cubic. But
and so
that are greater than
, where
is Lagrange's discriminant for the cubic, so
in the numerator until
is the Cardano discriminant of the
is the Lagrange discriminant of the auxiliary cubic. This is
always the case:
Let
be the roots of a given quartic and let
be the roots of the Ferrari auxiliary cubic. Then
Lagrange was interested in permutations of the roots of polynomials and how they combined with roots of unity. He proved several theorems
about these combinations (which came to be known as Lagrange resolvents) that led to the development of group theory. For the general cubic
with roots
, recall that
Lagrange defined the resolvents
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and
to be
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Since
is a primitive cube root of unity it follows that
and, after some tedious algebra,
But now, since
we can take
Then
can be written entirely in terms of the coefficients of the cubic:
We can take
to be any one of its cube roots, and then
, whose cube will of course satisfy the third equation of
. The
provide the roots of the cubic. In this way, Lagrange was able to reproduce Cardano's Formula with a consistent interpretation
equations of
of the cube root terms.
Note that if
For example,
then we can take
to be any cube root of
for the polynomial
for which
so we can take
as expected, since
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and obtain
. Then the roots of the cubic are
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so
.
Suggestions for Second Project:
, express the Lagrange discriminant
20. For the general cubic
not appear in the final form). Explain why
is independent of the coefficient
in terms of the resolvents
and
only (
will
.
Resolvents for Quadratics and Quartics
The resolvents
and
for the cubic were derived from a general theory introduced by Lagrange to study polynomials of any degree. If
are the roots of a polynomial of degree
some
and
is a primitive
root of unity (
and every root of unity is
for
) then
is an example of a Lagrange resolvent. Others can be obtained by replacing
have
with
for
as above. Thus, Lagrange was attempting to generalize the sum of the roots of a polynomial, which is
. For the cubic, with
(the coefficient of the trace
term), in hopes that these other resolvents could be expressed in terms of the coefficients of the original polynomial. Note that when
must take
, and then the resolvents are
Let
However,
and so the roots are
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. Then
, where
, we
is the constant term in the quadratic. Thus
we
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and we obtain the quadratic formula in a similar manner to the way Lagrange obtained Cardano's Formula from the cubic resolvents.
Lagrange then applied his theory to the general quartic
with
, a primitive fourth root of unity, forming the resolvents
Noticing that
and
, Lagrange found it easier to work with
because
and so
where
Lagrange now produced a resolvent cubic, similar but not identical to Ferrari's auxiliary cubic:
because
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These relations are tedious to find but Newton had developed powerful theorems for expressing symmetric polynomials of various degrees in
are computed in terms of the coefficients of the quartic by taking square roots.
terms of each other. After finding the roots of this cubic the
Since
it follows that
and so the square roots must be taken so as to satisfy this relation. As before, the roots of the quartic can be solved in terms of the
This method gives an interpretation to the resolvent cubic that previewed the Galois theory of polynomial roots. Note that if
resolvent cubic becomes
to obtain
then this
The Lagrange discriminant of this cubic is identical to the Lagrange discriminant of the original quartic, without having to introduce the factor
of
, because
From the viewpoint of group theory, Lagrange was exploiting the fact that
are each left invariant by a subgroup of order 4 of the
full symmetric group, yet are permuted among each other by the full symmetric group of order 24.
Note also that the discriminant
is easily expressed in terms of the
and so
whereas in terms of the original resolvents
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because
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Polynomials of Higher Degree
The method of Lagrange resolvents runs into difficulties for most polynomials of degree 5 or higher. Let
and suppose
be a primitive fifth root of unity.
Then the fundamental resolvents are
and so
For the cubic polynomial, the product of the resolvents was easily expressed in terms of the coefficients of the polynomial. For the quartic, the
product of the original resolvents was not a symmetric polynomial in the roots but Lagrange was able to replace those resolvents with linear
combinations of them so that the product became symmetric in the roots and hence expressible in terms of the coefficients of the quartic. For the
quintic we have
where
which is not symmetric and therefore cannot be expressed in terms of
. Further, there is no linear combination of the resolvents
whose product is symmetric. This was an indication to Lagrange that it is unlikely that powers of the individual resolvents could be expressed as
polynomials in the coefficients of the quintic. Thus, the roots of the general quintic might not be determinable by extraction of radicals.
In 1799, Paolo Ruffini (1765-1822) tried to prove the existence of quintics not solvable in radicals. In 1824, Niels Henrik Abel (1802-1829)
managed to do this. Of course, there are quintics whose roots can be determined by radicals. For example,
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and so the five roots are obtained by solving the quadratic and the cubic. Suppose, then, that the quintic is irreducible over , which means
that it does not factor into polynomials of lower degree with rational coefficients. Then the polynomial may or may not be solvable in radicals.
As a solvable example, consider
which has no rational root but which obviously has
where
as a root. (Here,
is a primitive fifth root of unity. Notice that
is the real fifth root of
but also
.) The other four roots are
in this case. However,
, consistent with the formula for the roots in terms of the resolvents.
If
is an integer not divisible by the square of any prime then
is irreducible over the rational numbers but is solvable in radicals.
These irreducible binomials have an important property: Any root is a rational function of any two of the roots. For example, if we let
and
for
, then
are the other three roots. Evariste Galois (1811-1832) discovered the following theorem that characterized solvable polynomials of any degree:
An irreducible polynomial of prime degree is solvable by radicals if and only if all roots are rational functions of any two of them.
For quintics there is an immediate corollary:
An irreducible quintic with three real roots and two non-real roots is not solvable by radicals.
Thus, a necessary (though not sufficient) condition for an irreducible quintic, or for any irreducible polynomial of prime degree
solvable by radicals is that it have all real roots or a unique real root. Consider the polynomial function
which is easily seen using calculus to have exactly three real zeros:
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, to be
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The polynomial
does not factor over the rational numbers. The other two roots guaranteed by the Fundamental Theorem of
Algebra are non-real complex conjugates. Since they cannot be produced as rational functions of the real roots this polynomial is not solvable by
radicals.
Radical Extensions of the Rational Numbers
The theory that Galois developed went much deeper than the statement of the above theorem. It described how the roots of solvable
polynomials could be obtained by extending the field of rational numbers . This amounted to a vast generalization of Lagrange's work on the
permutation of roots and explained how these permutations determined whether or not a polynomial is irreducible. The Fundamental Theorem of
Galois Theory establishes a correspondence between a particular group of permutations associated with a given irreducible polynomial and the
extensions of
where the roots of the polynomial are found. This theory requires a considerable amount of abstract algebra to describe, but
certain extensions of
equation. If
, called radical extensions, are not difficult to define. Examples of such extensions arose in the solution of Pell's
is an integer not divisible by the square of any prime then we can consider all numbers of the form
and denote this extension of
by
. Clearly the sum or product of any two such numbers is again such a number. For example,
However, the reciprocal of such a number is also in
For this reason,
is called a field extension of
these numbers. In this context we write
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because
because addition, subtraction, multiplication and division can be performed within
to emphasize this fact. If we wanted to factor
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this would require a field extension that contained the roots
. This extension does not contain the numbers
to create
and
and
. Such an extension is
but if we create the extension
and then extend by
then we get all numbers of the form
which contains the smaller extensions
, and
. These are all examples of what Galois called radical extensions
because they could be obtained from number fields by adjoining numbers which would be in the smaller field if a large enough integer power
were taken. Galois proved that a root of an irreducible polynomial belongs to a radical extension of
if and only if all of its roots belong to
some radical extension. He then showed that this condition is equivalent to being able to solve the polynomial by radicals. The polynomial
, then, has no root that belongs to a radical extension of the rational numbers. Nonetheless, we can take a root of this
polynomial and use it to describe the field extension of
where all of the roots belong. Starting with
we find that the reciprocal of
of the form
can be written as a polynomial in
for some rational numbers
. We say that the field
. Similarly, it can be shown that all numbers in the extension
is a vector space of dimension 5 over the field
. In general, if
are
is a
root of an polynomial of degree that is irreducible over
then
is a vector space of dimension over the field . Complex numbers
that occur as roots of polynomials with rational coefficients are called algebraic numbers. They comprise a countable set and are themselves a
field, which we refer to as the algebraic closure of .
Final Exam Topics
This is a test where you should use your own notes and be able to:
Find a formula for
in terms of the roots of a quadratic polynomial determined by the recursion
Use the discriminant to determine the nature of the roots of a cubic polynomial and find those roots using Cardano's Formula.
Find the roots of a quartic polynomial using Ferrari's biquadratic method.
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Find the roots of a quartic polynomial using Lagrange resolvents.
Express an element of
as a polynomial in
with rational coefficients.
Show that an irreducible quintic may not be solvable by radicals.
Eisenstein Criterion: The polynomial
such that
Solving a quartic with resolvents: For the quartic
cubic is
Using the resolvents
we have
Choosing
satisfies
Then
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, where the
, we have
are integers, is irreducible if there is a prime
, so the Lagrange's resolvent
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after using the identities
Note that
(Using Ferrari's biquadratic method, the auxiliary cubic is
whose roots are
. Selecting
Choosing the plus (minus) sign we obtain
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(
produces the two quadratics
) and its complex conjugate.)