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Algebraic closure

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Title: Algebraic closure  
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Subject: Triangular decomposition, Carlitz exponential, Irreducibility (mathematics), System of polynomial equations, Zorn's lemma
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Algebraic closure

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, it can be shown that every field has an algebraic closure,[1][2][3] and that the algebraic closure of a field K is unique up to an isomorphism that fixes every member of K. Because of this essential uniqueness, we often speak of the algebraic closure of K, rather than an algebraic closure of K.

The algebraic closure of a field K can be thought of as the largest algebraic extension of K. To see this, note that if L is any algebraic extension of K, then the algebraic closure of L is also an algebraic closure of K, and so L is contained within the algebraic closure of K. The algebraic closure of K is also the smallest algebraically closed field containing K, because if M is any algebraically closed field containing K, then the elements of M that are algebraic over K form an algebraic closure of K.

The algebraic closure of a field K has the same cardinality as K if K is infinite, and is countably infinite if K is finite.[3]


  • There are many countable algebraically closed fields within the complex numbers, and strictly containing the field of algebraic numbers; these are the algebraic closures of transcendental extensions of the rational numbers, e.g. the algebraic closure of Q(π).

Existence of an algebraic closure and splitting fields

Let S = \{ f_{\lambda} | \lambda \in \Lambda\} be the set of all monic irreducible polynomials in K[x]. For each f_{\lambda} \in S, introduce new variables u_{\lambda,1},\ldots,u_{\lambda,d} where d = {\rm degree}(f_{\lambda}). Let R be the polynomial ring over K generated by u_{\lambda,i} for all \lambda \in \Lambda and all i \leq {\rm degree}(f_{\lambda}). Write

f_{\lambda} - \prod_{i=1}^d (x-u_{\lambda,i}) = \sum_{j=0}^{d-1} r_{\lambda,j} \cdot x^j \in R[x]

with r_{\lambda,j} \in R. Let I be the ideal in R generated by the r_{\lambda,j}. By Zorn's lemma, there exists a maximal ideal M in R that contains I. Now R/M is an algebraic closure of K; every f_{\lambda} splits as the product of the x-(u_{\lambda,i} + M).

The same proof also shows that for any subset S of K[x], there exists a splitting field of S over K.

Separable closure

An algebraic closure Kalg of K contains a unique separable extension Ksep of K containing all (algebraic) separable extensions of K within Kalg. This subextension is called a separable closure of K. Since a separable extension of a separable extension is again separable, there are no finite separable extensions of Ksep, of degree > 1. Saying this another way, K is contained in a separably-closed algebraic extension field. It is essentially unique (up to isomorphism).[5]

The separable closure is the full algebraic closure if and only if K is a perfect field. For example, if K is a field of characteristic p and if X is transcendental over K, K(X)(\sqrt[p]{X}) \supset K(X) is a non-separable algebraic field extension.

In general, the absolute Galois group of K is the Galois group of Ksep over K.[6]

See also


  1. ^ McCarthy (1991) p.21
  2. ^ M. F. Atiyah and I. G. Macdonald (1969). Introduction to commutative algebra. Addison-Wesley publishing Company. pp. 11-12.
  3. ^ a b Kaplansky (1972) pp.74-76
  4. ^ .
  5. ^ McCarthy (1991) p.22
  6. ^
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