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theory of algebraic and transcendental
numbers∗
alozano†
2013-03-21 19:08:03
The following entry is some sort of index of articles in PlanetMath about the
basic theory of algebraic and transcendental numbers, and it should be studied
together with its complement: the theory of rational and irrational numbers.
The reader should follow the links in each bullet-point to learn more about
each topic. For a somewhat deeper approach to the subject, the reader should
read about Algebraic Number Theory. In this entry we will concentrate on the
properties of the complex numbers and the extension C/Q, however, in general,
one can talk about numbers of any field F which are algebraic over a subfield
K.
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Basic Definitions
1. A number α ∈ C is said to be algebraic (over Q), or an algebraic number,
if there is a polynomial p(x) with integer coefficients such that α is a root
of p(x) (i.e. p(α) = 0).
2. Similarly as the rational numbers may be classified to integer and noninteger (fractional) numbers, also the algebraic numbers may be classified
to algebraic integers or entire algebraic numbers and non-integer algebraic
numbers. The algebraic integers form an integral domain.
√ √ √
√
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3. The numbers −12, 2, 3 7, 2 +
7, ζ7 = e2πi/7 (that is, a 7th root of
√
unity), are all algebraic integers, 22 is a non-integer algebraic number (its
minimal polynomial is 2x2 − 1). See also rational algebraic integers.
4. A number α ∈ C is said to be transcendental if it is not algebraic.
5. For example, e is transcendental, where e is the natural log base (also
called the Euler number). The number Pi (π) is also transcendental. The
proofs of these two facts are HARD!
∗ hTheoryOfAlgebraicAndTranscendentalNumbersi
created: h2013-03-21i by: halozanoi
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6. A field extension L/K is said to be an algebraic extension if every element
of L is algebraic over K. An extension
√ which is not algebraic is said to
be transcendental. For example Q( 2)/Q is algebraic while Q(e)/Q is
transcendental (see the simple field extensions).
7. The algebraic closure of a field Q is the union of all algebraic extension
fields L of Q. The algebraic closure of Q is usually denoted by Q. In other
words, Q is the union of all complex numbers which are algebraic.
8. The set Q of all algebraic numbers is a field. It has as a subfield the
Q ∩ R, the set of all real algebraic numbers, and as a subring the set of
all algebraic integers. See the field of algebraic numbers and the ring of
algebraic integers.
9. The ring of all algebraic integers A contains no irreducible elements.
10. The height of an algebraic number is a way to measure the complexity of
the number.
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Small Results
1. A finite extension of fields is an algebraic extension.
2. The extension R/Q is not finite.
3. For every algebraic number α, there exists an irreducible minimal polynomial mα (x) such that mα (α) = 0 (see existence of the minimal polynomial).
4. For any algebraic number α, there is a nonzero multiple nα which is an
algebraic integer (see multiples of an algebraic number):
5. Some examples of algebraic numbers are the sine, cosine and tangent of
the angles rπ where r is a rational number (see this entry). More usual
are the root expressions of rational numbers.
6. The transcendental root theorem: Let F ⊂ K be a field extension with K
an algebraically closed field. Let x ∈ K be transcendental over F . Then
for any natural number n ≥ 1, the element x1/n ∈ K is also transcendental
over F .
7. An example of transcendental number (as an application of Liouville’s
approximation theorem).
8. The algebraic numbers are countable. In other words, Q is a countable
subset of C. Since C is uncountable, we conclude that there are infinitely
many transcendental numbers (uncountably many!). See also the proof
of the existence of transcendental numbers.
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9. Algebraic and transcendental: the sum, difference, and quotient of two
non-zero complex numbers, from which one is algebraic and the other
transcendental, is transcendental.
10. All transcendental extension fields Q(α) of Q are isomorphic (see the simple transcendental field extensions).
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BIG Results
1. Steinitz Theorem: There exists an algebraic closure of a field.
2. The Gelfond-Schneider Theorem: Let α and β be algebraic over Q, with
β irrational and α not equal to 0 or 1. Then αβ is transcendental over Q.
3. The Lindemann-Weierstrass Theorem.
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