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 bulletpoint 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^{} $\u2102/\mathbb{Q}$, however, in general, one can talk about numbers of any field $F$ which are algebraic over a subfield^{} $K$.
1 Basic Definitions

1.
A number $\alpha \in \u2102$ is said to be algebraic^{} (http://planetmath.org/Algebraic) (over $\mathbb{Q}$), or an algebraic number^{}, if there is a polynomial^{} $p(x)$ with integer coefficients such that $\alpha $ is a root of $p(x)$ (i.e. $p(\alpha )=0$).

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 algebraic numbers and noninteger algebraic numbers. The algebraic integers form an integral domain^{}.

3.
The numbers $12$, $\sqrt{2}$, $\sqrt[3]{7}$, $\sqrt{2}+\sqrt[3]{7}$, ${\zeta}_{7}={e}^{2\pi i/7}$ (that is, a $7$th root of unity^{}), are all algebraic integers, $\frac{\sqrt{2}}{2}$ is a noninteger algebraic number (its is $2{x}^{2}1$). See also rational algebraic integers.

4.
A number $\alpha \in \u2102$ is said to be transcendental if it is not algebraic.

5.
For example, e is transcendental, where e is the natural $\mathrm{log}$ base (also called the Euler number). The number Pi ($\pi $) is also transcendental. The proofs of these two facts are HARD!

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 $\mathbb{Q}(\sqrt{2})/\mathbb{Q}$ is algebraic while $\mathbb{Q}(e)/\mathbb{Q}$ is transcendental (see the simple field extensions).

7.
The algebraic closure^{} of a field $\mathbb{Q}$ is the union of all algebraic extension fields $L$ of $\mathbb{Q}$. The algebraic closure of $\mathbb{Q}$ is usually denoted by $\overline{\mathbb{Q}}$. In other words, $\overline{\mathbb{Q}}$ is the union of all complex numbers which are algebraic.

8.
The set $\overline{\mathbb{Q}}$ of all algebraic numbers is a field. It has as a subfield the $\overline{\mathbb{Q}}\cap \mathbb{R}$, 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 $\mathbb{A}$ contains no irreducible elements^{} (http://planetmath.org/RingWithoutIrreducibles).

10.
The height of an algebraic number is a way to measure the complexity of the number.
2 Small Results

1.
A finite extension^{} of fields is an algebraic extension.

2.
The extension $\mathbb{R}/\mathbb{Q}$ is not finite (http://planetmath.org/ExtensionMathbbRmathbbQIsNotFinite).

3.
For every algebraic number $\alpha $, there exists an irreducible minimal polynomial ${m}_{\alpha}(x)$ such that ${m}_{\alpha}(\alpha )=0$ (see existence of the minimal polynomial).

4.
For any algebraic number $\alpha $, there is a nonzero multiple^{} $n\alpha $ 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 $r\pi $ where $r$ is a rational number (see this entry (http://planetmath.org/AlgebraicSinesAndCosines)). More usual are the root expressions of rational numbers.

6.
The transcendental root theorem (http://planetmath.org/ProofOfTranscendentalRootTheorem): Let $F\subset K$ be a field extension with $K$ an algebraically closed field. Let $x\in K$ be transcendental over $F$. Then for any natural number^{} $n\ge 1$, the element ${x}^{1/n}\in K$ 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, $\overline{\mathbb{Q}}$ is a countable^{} subset of $\u2102$. Since $\u2102$ is uncountable, we conclude that there are infinitely many transcendental numbers^{} (uncountably many!). See also the proof of the existence of transcendental numbers.

9.
Algebraic and transcendental: the sum, difference^{}, and quotient of two nonzero complex numbers, from which one is algebraic and the other transcendental, is transcendental.

10.
All transcendental extension fields $\mathbb{Q}(\alpha )$ of $\mathbb{Q}$ are isomorphic (see the simple transcendental field extensions).
3 BIG Results

1.
Steinitz Theorem: There exists an algebraic closure of a field.

2.
The GelfondSchneider Theorem^{}: Let $\alpha $ and $\beta $ be algebraic over $\mathbb{Q}$, with $\beta $ irrational and $\alpha $ not equal to 0 or 1. Then ${\alpha}^{\beta}$ is transcendental over $\mathbb{Q}$.

3.
The LindemannWeierstrass Theorem^{}.
Title  theory of algebraic and transcendental numbers 
Canonical name  TheoryOfAlgebraicAndTranscendentalNumbers 
Date of creation  20130322 15:14:01 
Last modified on  20130322 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 