Algebraic element

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The terms algebraic and transcendent element appear in abstract algebra and generalize the concept of algebraic and transcendent numbers .

definition

Be an extension of the body , an element. Then is called algebraic about if there is a polynomial different from the zero polynomial with coefficients in that has as a zero.

An element from that is not algebraically over is called transcendent over .

Examples

  • A complex number is an algebraic number if and only if it is an algebraic element in the field extension .
  • The square root of 2 is algebraic over because it is a zero of the polynomial whose coefficients are rational.
  • The circle number and Euler's number are transcendent over . But they are algebraically over because they are defined as real numbers. More generally applies:
  • Every element of the field is algebraically about because it is the root of the linear polynomial .
  • Every complex number that can be formed by rational numbers, the basic arithmetic operations ( addition , subtraction , multiplication and division ) as well as by taking the root (with natural root exponents) is algebraic about .
  • However, it follows from Galois theory that, conversely, there are algebraic numbers that cannot be represented in this way; compare Abel-Ruffini's theorem .
  • Over the field of p-adic numbers is algebraically (as limit the number of reciprocal faculties), because for is and is in included.
  • If one forms the body of the formal Laurentreihen for any body , the formal variable is a transcendent element of this extension.

properties

The following conditions are equivalent for an element made of (an upper body of ):

  • is algebraically about .
  • The body expansion is of finite degree , i. i.e., is finite-dimensional as a vector space.

It is the Ringadjunktion of at which all the elements of is that as a polynomial over can write. is its quotient field in and consists of all elements of which can be written as with polynomials and over ( not equal to the zero polynomial).

This characterization can be used to show that the sum, difference, product and quotient of over algebraic elements are again algebraically over . The set of all over algebraic elements of forms an intermediate field of the extension , the so-called algebraic closure in . This term should not be confused with the algebraic closure of .

Minimal polynomial

If algebraic is over , then there is exactly one normalized polynomial from with the smallest degree and zero , this is called “the minimum polynomial from over ”. It is also called an algebraic element of degree relative to . has a possible basis as a vector space over the dimension . So the degree of expansion of is also the same .

example

is an algebraic element of degree 4 above , for from

the minimal polynomial results

,

thus a 4th degree polynomial. So there is a basis of as a vector space over . Another possible basis is , i. H.,

and is a degree 4 body extension.

generalization

The concept of the whole element can be defined in ring extensions . If one understands a body expansion as a ring expansion, then an element is there exactly if it is an algebraic element of the body expansion.

Individual evidence

  1. ^ Kurt Meyberg: Algebra II. Carl Hanser Verlag (1976), ISBN 3-446-12172-2 , definition 6.2.10.
  2. ^ Kurt Meyberg: Algebra II. Carl Hanser Verlag (1976), ISBN 3-446-12172-2 , sentence 6.3.3 and sentence 6.3.4.
  3. Kurt Meyberg: Algebra II. Carl Hanser Verlag (1976), ISBN 3-446-12172-2 , chapter 6.3.