theory of algebraic and transcendental numbers


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 TheoryMathworldPlanetmath. In this entry we will concentrate on the properties of the complex numbersMathworldPlanetmathPlanetmath and the extensionPlanetmathPlanetmathPlanetmath /, however, in general, one can talk about numbers of any field F which are algebraic over a subfieldMathworldPlanetmath K.

1 Basic Definitions

  1. 1.

    A number α is said to be algebraicMathworldPlanetmath (http://planetmath.org/Algebraic) (over ), or an algebraic numberMathworldPlanetmath, if there is a polynomialPlanetmathPlanetmath p(x) with integer coefficients such that α is a root of p(x) (i.e. p(α)=0).

  2. 2.

    Similarly as the rational numbers may be classified to integer and non-integer (fractional) numbers, also the algebraic numbers may be classified to algebraic integersMathworldPlanetmath or algebraic numbers and non-integer algebraic numbers.  The algebraic integers form an integral domainMathworldPlanetmath.

  3. 3.

    The numbers -12, 2, 73, 2+73, ζ7=e2πi/7 (that is, a 7th root of unityMathworldPlanetmath), are all algebraic integers, 22 is a non-integer algebraic number (its is 2x2-1).  See also rational algebraic integers.

  4. 4.

    A number α is said to be transcendental if it is not algebraic.

  5. 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!

  6. 6.

    A field extension L/K is said to be an algebraic extensionMathworldPlanetmath if every element of L is algebraic over K. An extension which is not algebraic is said to be transcendental. For example (2)/ is algebraic while (e)/ is transcendental (see the simple field extensions).

  7. 7.

    The algebraic closureMathworldPlanetmath of a field is the union of all algebraic extension fields L of . The algebraic closure of is usually denoted by ¯. In other words, ¯ is the union of all complex numbers which are algebraic.

  8. 8.

    The set ¯ of all algebraic numbers is a field.  It has as a subfield the ¯, 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. 9.

    The ring of all algebraic integers 𝔸 contains no irreducible elementsMathworldPlanetmath (http://planetmath.org/RingWithoutIrreducibles).

  10. 10.

    The height of an algebraic number is a way to measure the complexity of the number.

2 Small Results

  1. 1.

    A finite extensionMathworldPlanetmath of fields is an algebraic extension.

  2. 2.

    The extension / is not finite (http://planetmath.org/ExtensionMathbbRmathbbQIsNotFinite).

  3. 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. 4.

    For any algebraic number α, there is a nonzero multipleMathworldPlanetmath nα which is an algebraic integer (see multiples of an algebraic number):

  5. 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 (http://planetmath.org/AlgebraicSinesAndCosines)).  More usual are the root expressions of rational numbers.

  6. 6.

    The transcendental root theorem (http://planetmath.org/ProofOfTranscendentalRootTheorem): Let FK be a field extension with K an algebraically closed field. Let xK be transcendental over F. Then for any natural numberMathworldPlanetmath n1, the element x1/nK is also transcendental over F.

  7. 7.

    An example of transcendental number (as an application of Liouville’s approximation theorem).

  8. 8.

    The algebraic numbers are countable. In other words, ¯ is a countableMathworldPlanetmath subset of . Since is uncountable, we conclude that there are infinitely many transcendental numbersMathworldPlanetmath (uncountably many!).  See also the proof of the existence of transcendental numbers.

  9. 9.

    Algebraic and transcendental:  the sum, differencePlanetmathPlanetmath, and quotient of two non-zero complex numbers, from which one is algebraic and the other transcendental, is transcendental.

  10. 10.

    All transcendental extension fields (α) of are isomorphic (see the simple transcendental field extensions).

3 BIG Results

  1. 1.

    Steinitz Theorem: There exists an algebraic closure of a field.

  2. 2.

    The Gelfond-Schneider TheoremMathworldPlanetmath: Let α and β be algebraic over , with β irrational and α not equal to 0 or 1. Then αβ is transcendental over .

  3. 3.
Title theory of algebraic and transcendental numbers
Canonical name TheoryOfAlgebraicAndTranscendentalNumbers
Date of creation 2013-03-22 15:14:01
Last modified on 2013-03-22 15:14:01
Owner alozano (2414)
Last modified by alozano (2414)
Numerical id 32
Author alozano (2414)
Entry type Topic
Classification msc 11R04
Related topic AlgebraicNumberTheory
Related topic TheoryOfRationalAndIrrationalNumbers
Related topic MultiplesOfAnAlgebraicNumber
Related topic NormAndTraceOfAlgebraicNumber
Related topic AlgebraicSumAndProduct
Related topic DegreeOfAnAlgebraicNumber