Algebraic element

The terms of algebraic and transcendental element occur in abstract algebra and generalize the concept of algebraic and transcendental numbers.

L / K, a body extension, it means an element a of L algebraically over K, if there is a different from the zero polynomial polynomial with coefficients in K, which has as a zero point. An item for which there is no such polynomial is, transcendental over K.

For the extension of these concepts are consistent with those of algebraic or transcendental number.

Examples

• The square root of 2 is algebraic over, because it is a root of the polynomial whose coefficients are rational.
• The county number and the Euler number is transcendental over, but algebraic over, because they are defined as real numbers. More generally,
• Each element a of the body K is algebraically to K, because there is zero point of the linear polynomial.
• Each complex number that can be formed by rational numbers, basic arithmetic operations (addition, subtraction, multiplication and division), as well as by root extraction ( with natural exponents of the root ) is algebraic over.
• However, from the Galois theory follows the other hand that there are over algebraic numbers that can not be represented in this way; cf the theorem of Abel - Ruffini.
• Over the field of p- adic numbers e is algebraic ( as a limit of the series of reciprocal faculties), because for p> 2 and for p = 2 is included.
• If one forms at any field K the body of formal Laurent series, the formal variable X is a transcendent element of this extension.

Properties

The following conditions are equivalent for an element a of L ( upper torso of K):

• A is algebraic over K
• The field extension has finite degree, that is, is finite as K- vector space.

Here is the Ringadjunktion of a of K, consisting of all elements of L, which can be as writing with a polynomial g over K; is the quotient field L and consists of all elements of L, which can be written as polynomials g and h over K ( equal to 0).

This characterization can be used to show that the sum, difference, product and the quotient of the algebraic over K elements are algebraically K again. The set of all algebraic over K elements of L form an intermediate body of the extension, the algebraic degree as referred to in L.

Minimal polynomial

If a is algebraic over K, then there is with many polynomials. But there is exactly one monic polynomial of least degree with zero a, this is called the minimal polynomial of a over K. From it one can read many of the properties of a. For example, the degree of this polynomial is equal to the degree of expansion.