Algebraic closure (original) (raw)

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Algebraic field extension

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 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]

Existence of an algebraic closure and splitting fields

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Let S = { f λ | λ ∈ Λ } {\displaystyle S=\{f_{\lambda }|\lambda \in \Lambda \}} {\displaystyle S=\{f_{\lambda }|\lambda \in \Lambda \}} be the set of all monic irreducible polynomials in _K_[_x_]. For each f λ ∈ S {\displaystyle f_{\lambda }\in S} {\displaystyle f_{\lambda }\in S}, introduce new variables u λ , 1 , … , u λ , d {\displaystyle u_{\lambda ,1},\ldots ,u_{\lambda ,d}} {\displaystyle u_{\lambda ,1},\ldots ,u_{\lambda ,d}} where d = d e g r e e ( f λ ) {\displaystyle d={\rm {degree}}(f_{\lambda })} {\displaystyle d={\rm {degree}}(f_{\lambda })}. Let R be the polynomial ring over K generated by u λ , i {\displaystyle u_{\lambda ,i}} {\displaystyle u_{\lambda ,i}} for all λ ∈ Λ {\displaystyle \lambda \in \Lambda } {\displaystyle \lambda \in \Lambda } and all i ≤ d e g r e e ( f λ ) {\displaystyle i\leq {\rm {degree}}(f_{\lambda })} {\displaystyle i\leq {\rm {degree}}(f_{\lambda })}. Write

f λ − ∏ i = 1 d ( x − u λ , i ) = ∑ j = 0 d − 1 r λ , j ⋅ x j ∈ R [ x ] {\displaystyle f_{\lambda }-\prod _{i=1}^{d}(x-u_{\lambda ,i})=\sum _{j=0}^{d-1}r_{\lambda ,j}\cdot x^{j}\in R[x]} {\displaystyle 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 λ , j ∈ R {\displaystyle r_{\lambda ,j}\in R} {\displaystyle r_{\lambda ,j}\in R}. Let I be the ideal in R generated by the r λ , j {\displaystyle r_{\lambda ,j}} {\displaystyle r_{\lambda ,j}}. Since I is strictly smaller than R, Zorn's lemma implies that there exists a maximal ideal M in R that contains I. The field _K_1=R/M has the property that every polynomial f λ {\displaystyle f_{\lambda }} {\displaystyle f_{\lambda }} with coefficients in K splits as the product of x − ( u λ , i + M ) , {\displaystyle x-(u_{\lambda ,i}+M),} {\displaystyle x-(u_{\lambda ,i}+M),} and hence has all roots in _K_1. In the same way, an extension _K_2 of _K_1 can be constructed, etc. The union of all these extensions is the algebraic closure of K, because any polynomial with coefficients in this new field has its coefficients in some K n with sufficiently large n, and then its roots are in K n+1, and hence in the union itself.

It can be shown along the same lines that for any subset S of _K_[_x_], there exists a splitting field of S over K.

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 unique (up to isomorphism).[7]

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 ) ( X p ) ⊃ K ( X ) {\displaystyle K(X)({\sqrt[{p}]{X}})\supset K(X)} {\displaystyle 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.[8]

  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. ^ Banaschewski, Bernhard (1992), "Algebraic closure without choice.", Z. Math. Logik Grundlagen Math., 38 (4): 383–385, doi:10.1002/malq.19920380136, Zbl 0739.03027
  5. ^ Mathoverflow discussion
  6. ^ Brawley, Joel V.; Schnibben, George E. (1989), "2.2 The Algebraic Closure of a Finite Field", Infinite Algebraic Extensions of Finite Fields, Contemporary Mathematics, vol. 95, American Mathematical Society, pp. 22–23, ISBN 978-0-8218-5428-0, Zbl 0674.12009.
  7. ^ McCarthy (1991) p.22
  8. ^ Fried, Michael D.; Jarden, Moshe (2008). Field arithmetic. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. Vol. 11 (3rd ed.). Springer-Verlag. p. 12. ISBN 978-3-540-77269-9. Zbl 1145.12001.