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In mathematics, particularly abstract algebra, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed. It is one of many closures in mathematics. Using Zorn's lemma [ 1 ] [ 2 ] [ 3 ] or the weaker ultrafilter lemma , [ 4 ] [ 5 ] it can be shown that every field has an algebraic closure , and that the ...
In the context of algebraic structures, this closure is generally called the substructure generated or spanned by X, and one says that X is a generating set of the substructure. For example, a group is a set with an associative operation, often called multiplication, with an identity element, such that every element has an inverse element.
The field F is algebraically closed if and only if it has no proper algebraic extension. If F has no proper algebraic extension, let p(x) be some irreducible polynomial in F[x]. Then the quotient of F[x] modulo the ideal generated by p(x) is an algebraic extension of F whose degree is equal to the degree of p(x).
For example, the real closure of the ordered field of rational numbers is the field of real algebraic numbers. The theorem is named for Emil Artin and Otto Schreier , who proved it in 1926. If ( F , P ) is an ordered field, and E is a Galois extension of F , then by Zorn's lemma there is a maximal ordered field extension ( M , Q ) with M a ...
Imperfect fields cause technical difficulties because irreducible polynomials can become reducible in the algebraic closure of the base field. For example, [4] consider (,) = + [,] for an imperfect field of characteristic and a not a p-th power in k. Then in its algebraic closure [,], the following equality holds:
It is commonly referred to as the algebraic closure and denoted F. For example, the algebraic closure Q of Q is called the field of algebraic numbers. The field F is usually rather implicit since its construction requires the ultrafilter lemma, a set-theoretic axiom that is weaker than the axiom of choice. [34] In this regard, the algebraic ...
Alternatively it is the group of all automorphisms of the algebraic closure of K that fix K. The absolute Galois group is well-defined up to inner automorphism. It is a profinite group. (When K is a perfect field, K sep is the same as an algebraic closure K alg of K. This holds e.g. for K of characteristic zero, or K a finite field.)
Convex hull (red) of a polygon (yellow). The usual set closure from topology is a closure operator. Other examples include the linear span of a subset of a vector space, the convex hull or affine hull of a subset of a vector space or the lower semicontinuous hull ¯ of a function : {}, where is e.g. a normed space, defined implicitly (¯) = ¯, where is the epigraph of a function .