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 $\Complex/\Rats$ , however, in general, one can talk about numbers of any field $F$ which are algebraic over a subfield $K$ .

Basic Definitions

1. A number $\alpha\in \Complex$ is said to be algebraic (over $\Rats$ ), 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 non-integer (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.
3. The numbers $\sqrt{2}$ , $\sqrt[3]{7}$ , $\sqrt{2}+\sqrt[3]{7}$ , $\zeta_{7}=e^{2\pi i/7}$ (that is, a $7$ th root of unity), are all algebraic integers, $\frac{\sqrt{2}}{2}$ is a non-integer algebraic number (its minimal polynomial is $2x^2-1$ ).
4. A number $\alpha\in \Complex$ 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 ($\pi$ ) 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 $\Rats(\sqrt{2})/\Rats$ is algebraic while $\Rats(e)/\Rats$ is transcendental (see the simple field extensions).
7. The algebraic closure of a field $\Rats$ is the union of all algebraic extension fields $L$ of $\Rats$ . The algebraic closure of $\Rats$ is usually denoted by $\overline{\Rats}$ . In other words, $\overline{\Rats}$ is the union of all complex numbers which are algebraic.
8. The set $\overline{\Rats}$ of all algebraic numbers is a field. It has as a subfield the $\overline{\Rats}\cap\Reals$ , the set of all real algebraic numbers, and as a subring the set of all algebraic integers.
9. The ring of all algebraic integers $\mathbb{A}$ contains no irreducible elements.
10. The height of an algebraic number is a way to measure the complexity of the number.

Small Results

1. A finite extension of fields is an algebraic extension.
2. The extension $\Reals/\Rats$ is not finite.
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. 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). More usual are the root expressions of rational numbers.
5. The transcendental root theorem: 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 $n\geq 1$ , the element $x^{1/n}\in K$ is also transcendental over $F$ .
6. An example of transcendental number (as an application of Liouville's approximation theorem).
7. The algebraic numbers are countable. In other words, $\overline{\Rats}$ is a countable subset of $\Complex$ . Since $\Complex$ is uncountable, we conclude that there are infinitely many transcendental numbers (uncountably many!). See also the proof of the existence of transcendental numbers.
8. 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.
9. All transcendental extension fields $\Rats(\alpha)$ of $\Rats$ are isomorphic (see the simple transcendental field extensions).

BIG Results

1. Steinitz Theorem: There exists an algebraic closure of a field.
2. The Gelfond-Schneider 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 Lindemann-Weierstrass Theorem.