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 , however, in general, one can talk about numbers of any field which are algebraic over a subfield .
1 Basic Definitions
A number is said to be transcendental if it is not algebraic.
For example, e is transcendental, where e is the natural base (also called the Euler number). The number Pi () is also transcendental. The proofs of these two facts are HARD!
The algebraic closure of a field is the union of all algebraic extension fields of . The algebraic closure of is usually denoted by . In other words, is the union of all complex numbers which are algebraic.
The ring of all algebraic integers contains no irreducible elements (http://planetmath.org/RingWithoutIrreducibles).
The height of an algebraic number is a way to measure the complexity of the number.
2 Small Results
A finite extension of fields is an algebraic extension.
The extension is not finite (http://planetmath.org/ExtensionMathbbRmathbbQIsNotFinite).
For any algebraic number , there is a nonzero multiple which is an algebraic integer (see multiples of an algebraic number):
Some examples of algebraic numbers are the sine, cosine and tangent of the angles where is a rational number (see this entry (http://planetmath.org/AlgebraicSinesAndCosines)). More usual are the root expressions of rational numbers.
3 BIG Results
Steinitz Theorem: There exists an algebraic closure of a field.
|Title||theory of algebraic and transcendental numbers|
|Date of creation||2013-03-22 15:14:01|
|Last modified on||2013-03-22 15:14:01|
|Last modified by||alozano (2414)|