Download Why Abstract Algebra? (Outline of Chapter 1 of

Why Abstract Algebra? (Outline of Chapter 1 of
Pinter’s Book)
Sean Ellermeyer
Kennesaw State University
August 17, 2015
Sean Ellermeyer (Kennesaw State University)Why Abstract Algebra? (Outline of Chapter 1 of Pinter’s Book)
August 17, 2015
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The word “algebra” comes from “al jabr” in Arabic. The rough translation
of this word is “reunion”. This refers to the collecting of like terms to
solve equations. The word was first coined by the mathematician Abu
Ja’far Muhammad ibn Musa Al-Khwarizmi (790 – 850). Omar Khayyam
(1048–1131) defined “algebra” to be “the science of solving equations”.
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Linear and Quadratic Equations
Procedures (formulas) for solving linear equations
ax + b = 0, a 6= 0
and quadratic equations
ax 2 + bx + c = 0, a 6= 0
were known since ancient times (for instance, in ancient Greek society) but
no such procedure was known for solving cubic equations
ax 3 + bx 2 + cx + d = 0, a 6= 0
or polynomial equations of any degree n > 3. To find such procedures was
a problem that occupied mathematicians for many centuries.
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Example of a Cubic Equation
An example of a cubic equation is
4x 3 + 12x 2 + x + 3 = 0.
The solutions of this equation are x = −3, x = 21 i and x = − 21 i (where i
is the complex number defined such that i 2 = −1). It is easily checked
that these numbers are solutions but how do we find them? Is there a
formula (along the same lines as the quadratic formula) that we can use to
solve equations of the form
ax 3 + bx 2 + cx + d = 0, a 6= 0?
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Developments During the Italian Renaissance
Girolamo Cardano (1501–1576). To quote from Charles Pinter,
Cardano was a physician, an astrologer, a mathematician, a
compulsive gambler, a scoundrel and a heretic.
He was initially not successful with his medical practice and he and his
wife had to seek refuge in a poorhouse.
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Developments During the Italian Renaissance
Girolamo Cardano (1501–1576). To quote from Charles Pinter,
Cardano was a physician, an astrologer, a mathematician, a
compulsive gambler, a scoundrel and a heretic.
He was initially not successful with his medical practice and he and his
wife had to seek refuge in a poorhouse.
He was eventually able to land a position as a lecturer of mathematics.
Sean Ellermeyer (Kennesaw State University)Why Abstract Algebra? (Outline of Chapter 1 of Pinter’s Book)
August 17, 2015
5 / 15
Developments During the Italian Renaissance
Girolamo Cardano (1501–1576). To quote from Charles Pinter,
Cardano was a physician, an astrologer, a mathematician, a
compulsive gambler, a scoundrel and a heretic.
He was initially not successful with his medical practice and he and his
wife had to seek refuge in a poorhouse.
He was eventually able to land a position as a lecturer of mathematics.
Later he was very successful in medicine and was the first physician to
give a clinical description of typhus fever.
Sean Ellermeyer (Kennesaw State University)Why Abstract Algebra? (Outline of Chapter 1 of Pinter’s Book)
August 17, 2015
5 / 15
Developments During the Italian Renaissance
Girolamo Cardano (1501–1576). To quote from Charles Pinter,
Cardano was a physician, an astrologer, a mathematician, a
compulsive gambler, a scoundrel and a heretic.
He was initially not successful with his medical practice and he and his
wife had to seek refuge in a poorhouse.
He was eventually able to land a position as a lecturer of mathematics.
Later he was very successful in medicine and was the first physician to
give a clinical description of typhus fever.
Due to his interest in gambling, he took a great interest in probability
and became an expert on the subject. He wrote the book Book on
Games of Chance.
Sean Ellermeyer (Kennesaw State University)Why Abstract Algebra? (Outline of Chapter 1 of Pinter’s Book)
August 17, 2015
5 / 15
Developments During the Italian Renaissance
Girolamo Cardano (1501–1576). To quote from Charles Pinter,
Cardano was a physician, an astrologer, a mathematician, a
compulsive gambler, a scoundrel and a heretic.
He was initially not successful with his medical practice and he and his
wife had to seek refuge in a poorhouse.
He was eventually able to land a position as a lecturer of mathematics.
Later he was very successful in medicine and was the first physician to
give a clinical description of typhus fever.
Due to his interest in gambling, he took a great interest in probability
and became an expert on the subject. He wrote the book Book on
Games of Chance.
He also wrote the famous Ars Magna which means “The Great Art”. It
was a book that compiled all that was known about algebra at the time.
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Niccolò Fontana Tartaglia (1500–1557)
He had no formal education. He was self–taught in mathematics.
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Niccolò Fontana Tartaglia (1500–1557)
He had no formal education. He was self–taught in mathematics.
He was the originator of the science of ballistics.
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Niccolò Fontana Tartaglia (1500–1557)
He had no formal education. He was self–taught in mathematics.
He was the originator of the science of ballistics.
In 1535 he discovered how to solve the cubic equation
ax 3 + bx 2 + d = 0 (which is a special case of the general cubic
equation ax 3 + bx 2 + cx + d = 0). As was customary at the time, he
would not share the secret of his method with other people.
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Scipione del Ferro (1465–1526)
He discovered how to solve the cubic equation ax 3 + cx + d = 0 but
kept his method a secret until, on his death bed, he revealed the secret
to his student Antonio Fior.
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August 17, 2015
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Scipione del Ferro (1465–1526)
He discovered how to solve the cubic equation ax 3 + cx + d = 0 but
kept his method a secret until, on his death bed, he revealed the secret
to his student Antonio Fior.
Armed with this new secret information, Fior challenged Tartaglia to an
algebra contest (for money). Each contestant was to create 30
problems for the other to solve and whoever solved the most problems
won the money.
Sean Ellermeyer (Kennesaw State University)Why Abstract Algebra? (Outline of Chapter 1 of Pinter’s Book)
August 17, 2015
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Scipione del Ferro (1465–1526)
He discovered how to solve the cubic equation ax 3 + cx + d = 0 but
kept his method a secret until, on his death bed, he revealed the secret
to his student Antonio Fior.
Armed with this new secret information, Fior challenged Tartaglia to an
algebra contest (for money). Each contestant was to create 30
problems for the other to solve and whoever solved the most problems
won the money.
A few days before the contest, Tartaglia discovered how to solve the
general cubic equation ax 3 + bx 2 + cx + d = 0 and he thus won the
contest easily. (Tartaglia solved all 30 problems that were given to him
and Fior solved none.)
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August 17, 2015
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Cardano and Tartaglia
Cardano badly wanted to learn Tartaglia’s method for solving cubic
equations. He persuaded Tartaglia to reveal his secret in exchange for
promising to help Tartaglia get a position as artillery advisor to the
Spanish army. Tartaglia taught his method to Cardano but swore him to
secrecy. Later Tartaglia was horrified to learn that Cardano had published
the method in Ars Magna. Although Cardano gave full credit to Tartaglia
for this major discovery, Tartaglia never forgave Cardano for breaking his
promise. Due to the feud that then started between the two, Tartaglia was
forced to live out the rest of his life in obscurity. In an unrelated matter,
Cardano was arrested for heresy for publishing a horoscope of the life of
Christ. He was jailed for several months, lost his university position and
was forbidden from publishing any more books.
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Ludovico Ferrari (1522-1565)
He was the personal assistant of Cardano and was taught mathematics
by Cardano.
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Ludovico Ferrari (1522-1565)
He was the personal assistant of Cardano and was taught mathematics
by Cardano.
In 1540, he discovered how to solve the quartic equation
ax 4 + bx 3 + cx 2 + dx + e = 0 given that the cubic equation could be
solved. Once the cubic equation had been solved (by Tartaglia),
Cardano was also able to include the method for solving the quartic
equation in Ars Magna which was published in 1545.
Sean Ellermeyer (Kennesaw State University)Why Abstract Algebra? (Outline of Chapter 1 of Pinter’s Book)
August 17, 2015
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Ludovico Ferrari (1522-1565)
He was the personal assistant of Cardano and was taught mathematics
by Cardano.
In 1540, he discovered how to solve the quartic equation
ax 4 + bx 3 + cx 2 + dx + e = 0 given that the cubic equation could be
solved. Once the cubic equation had been solved (by Tartaglia),
Cardano was also able to include the method for solving the quartic
equation in Ars Magna which was published in 1545.
Now that the methods were known for solving polynomial equations of
degree 1, 2, 3 and 4, the next problem to be tackled was how to solve
equation of degree higher than 4. This problem was studied by all of
the greatest mathematical minds during the next 300 years but no
solution was found.
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Neils Henrik Abel (1802–1829)
Abel lived his life in poverty and was not well–recognized during his
short lifetime. He died of tuberculosis at the age of 26.
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Neils Henrik Abel (1802–1829)
Abel lived his life in poverty and was not well–recognized during his
short lifetime. He died of tuberculosis at the age of 26.
Abel proved that there is no method for solving the quintic equation
ax 5 + bx 4 + cx 3 + dx 2 + ex + f = 0 or for solving polynomial
equations of any degree higher than 4! Specifically he proved that no
formula involving basic arithmetic operations of addition, subtraction,
multiplication, division and extraction of roots exists for solving
polynomial equations of degree higher than 4.
Sean Ellermeyer (Kennesaw State University)Why Abstract Algebra? (Outline of Chapter 1 of Pinter’s Book)
August 17, 2015
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Neils Henrik Abel (1802–1829)
Abel lived his life in poverty and was not well–recognized during his
short lifetime. He died of tuberculosis at the age of 26.
Abel proved that there is no method for solving the quintic equation
ax 5 + bx 4 + cx 3 + dx 2 + ex + f = 0 or for solving polynomial
equations of any degree higher than 4! Specifically he proved that no
formula involving basic arithmetic operations of addition, subtraction,
multiplication, division and extraction of roots exists for solving
polynomial equations of degree higher than 4.
As a tool needed to prove his result on the unsolvability of quintic
equations (and equations of higher degree), Abel invented the subject
of Group Theory (which was also invented independently by Galois).
The introduction of Group Theory led algebra in a completely new
direction with broader goals beyond just solving equations. This was
the beginning of the study of what is today called modern algebra or
abstract algebra.
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The Course of Abstract Algebra after Abel
Inspired by the great discoveries of Abel and Galois regarding the solutions
of polynomial equations and the corresponding discovery of group theory,
mathematicians began to view algebra in a different way and began to ask
different questions. The central problems of algebra were no longer just
questions about how to solve equations. The word “algebra” came to be
used more often as a noun to denote different kinds of algebraic
structures. For example, a group is a certain kind of algebra that is defined
by certain axioms. However there are many different algebras that are
defined by different axioms. The study of such algebras gradually arose as
they were encountered in trying to solve applied problems. We will
consider a few example of such algebras that you have already encountered
in your previous mathematics courses.
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The Algebra of Matrices
A matrix is a rectangular array of numbers

−4 7
8

−8 −5
7
A=
0
5 −10
such as

10 −6
6 −6  .
1
1
In linear algebra courses we define how to add two matrices of the same
size to obtain another matrix (of the same size). In fact, the set of all
m × n matrices (for some given m and n) is a group with the operation of
addition. However, if we restrict our attention only to the set of square
matrices (say 2 × 2 matrices), then we know that we can also define a very
useful multiplication operation on this set with which we multiply two
matrices of the same size and obtain another matrix (of the same size).
Thus we have an algebraic structure (or simply an “algebra”) that consists
of a set with two operations. One of the operations (addition) is
commutative and the other (multiplication) is not. Furthermore not every
square matrix has a multiplicative inverse so the set of square matrices is
not a group under multiplication.
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The Algebra of Vectors
A vector in R 3 is an ordered triple of numbers such as v = h3, −9, 7i. In
multivariable calculus courses we learn that we can define a useful addition
operation on R 3 . In fact R 3 is a group under vector addition. However we
also learn that we can define a useful “multiplication” operation called the
cross product. The cross product operation is denoted by the symbol ×.
You may recall that the cross product of two vectors in R 3 is another
vector in R 3 . You may also recall that the cross product is not
commutative. This reminds us of the situation that occurs with the
multiplication of matrices. For square matrices we generally have
AB 6= BA. Likewise with the cross product we have v × w 6= w × v.
However we can be more specific in the case of the cross product because
we do in fact know that the cross product operation is anti–commutative
meaning that v × w = − (w × v). Thus the the cross product operation is
more “specialized” in some sense than is the operation of matrix
multiplication.
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Vector Spaces
In linear algebra courses we are introduced to the algebraic structure called
a vector space. A vector space consists of two sets and two operations.
One set, V , is called the set of vectors and its operation is called
addition. The other set, F , is called the field of scalars. A field is an
algebraic structure in its own right which actually has two operations! In
introductory linear algebra courses we always assume that this field is the
field of real numbers (with the usual addition and multiplication). Besides
the addition operation, +, on V we define the scalar multiplication
operation (usually not denoted by any symbol) which is actually an
operation from F × V into V . That is, if k ∈ F and v ∈ V , then kv ∈ V .
There is a list of axioms that define exactly what we mean by a vector
space. For example we require that if k ∈ F and v and w ∈ V , then
k (v + w) = kv + kw.
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The General Idea
There are obviously limitless possibilities to the kinds of algebras that we
could study. In general we begin with
A set of sets {S1 , S2 , . . . , Sn }
A set of operations – some of which may be operations on a single of
the sets Si and some which may act between the sets
A set of axioms that tell us the rules which the operations must
satisfy.
This is the “modern” or “abstract” approach to algebra.
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