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 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 $-12$ , $\sqrt{2}$ , $\sqrt[3]{7}$ , $\sqrt{2}+\sqrt[3]{7}$ , $\zeta_{7}=e^{2\pi i/7}$ (that is, a $7$ th root of unity), are all algebraic integers, $\frac{\sqrt{2}}{2}$ is a non-integer algebraic number (its minimal polynomial is $2x^2-1$ ). See also rational algebraic integers.
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. 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.
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. 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). More usual are the root expressions of rational numbers.
6. 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 $F$ .
7. An example of transcendental number (as an application of Liouville's approximation theorem).
8. 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.
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 $\Rats(\alpha)$ 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.