Algebraic closure

A body is called algebraically closed if every nonconstant polynomial with coefficients in a zero in has. A body is an algebraic closure of, if it is algebraically closed and is an algebraic extension field of. Since an algebraic closure is unique up to isomorphism, one often speaks of the algebraic degree. The finding roots of polynomials is an important mathematical problem, in an algebraic degree can at least their existence be secured. In fact, one can show that there exists for every body an algebraic statements.

Definitions

Denote as usual over the polynomial ring.

General ie, a body algebraically closed if one of the following equivalent statements holds:

  • Every polynomial of has a zero in.
  • Every polynomial splits into linear factors from, so polynomials of degree 1
  • Has no real algebraic extensions.
  • Every irreducible polynomial has degree 1

An algebraic closure of a body can now be defined in two ways:

  • Is an algebraic extension field of in which every polynomial has a null point.
  • Is an algebraic extension field of in which every polynomial has a null point.

The second condition is a seemingly stronger statement, but it turns out to be equivalent to the first.

Existence

To a single polynomial, one can easily find an algebraic extension, in which the polynomial has a root. By the lemma of Zorn can find an algebraic extension in which all non-constant polynomials from a zero point. This is according to the above remark an algebraic closure of.

It succeeded Ernst Steinitz in 1910 as the first to show that every body has an algebraically closed upper body and thus an algebraic statements. This Steinitz use the axiom of choice, which is equivalent to the above-mentioned lemma of anger. The proof of the existence necessarily requires transfinite methods such as the axiom of choice: Are the axioms of set theory consistent, then so are the axioms of set theory together with the sentence "There is a body that has no algebraic degree. " consistent

Unambiguity

Also with the anger between lemma, one can show that two algebraic statements are mutually isomorphic, that is, for algebraic statements of there is a Körperisomorphismus, which is limited to the identity. However, there is no canonical, that is not excellent isomorphism, but in general very many equal. To be an algebraic closure, therefore, is not a universal property.

The algebraic closure of has the same cardinality as if is infinite, and is countable if is finite. An algebraically closed field can not be finite. Because the body is finite and p is the product of all the ( finitely many! ) Linear factors, then p 1 has no zeros.

Examples

  • The fundamental theorem of algebra states that the field of complex numbers is algebraically closed and hence is an algebraic degree of real numbers. If a different algebraic closure of and are and the solutions of in, so there are two isomorphisms from to. Either is mapped to or on. Both possibilities have equal rights.
  • An algebraic closure of the rational numbers, the field of algebraic numbers.
  • There are many countable algebraically closed real algebraic numbers in the upper body. You are algebraic statements transcendental extensions.
  • For a finite field of prime - order of the algebraic closure is a countably infinite field of characteristic, and contains, for each natural number a part of the body of the order, he is even from the union of this body part.

Importance

The significance of the algebraic statements is to find the zeros of polynomials. In algebraic degree has any polynomial of degree exactly zeros, which are to be counted with multiplicities. It, however, nothing is said about how these are to find concrete, see the article root.

Degree of a polynomial Zorn's lemma Characteristic (algebra)
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