- Algebraic closure
In

mathematics , particularlyabstract algebra , an**algebraic closure**of a field "K" is analgebraic 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, and that the algebraic closure of a field "K" is uniqueup to anisomorphism 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" which 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.**Examples***The

fundamental theorem of algebra states that the algebraic closure of the field ofreal number s is the field ofcomplex number s.*The algebraic closure of the field of

rational number s is the field ofalgebraic number s.*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).*For a

finite field of prime order "p", the algebraic closure is acountably infinite field which contains a copy of the field of order "p"^{"n"}for each positiveinteger "n" (and is in fact the union of these copies). (The most famous of these is the field ofNimbers , which is this field for "p" = 2)*See also

Puiseux expansion .**eparable closure**An algebraic closure "K

^{alg}" of "K" contains a uniqueseparable extension "K^{sep}" of "K" containing all (algebraic)separable extension s of "K" within "K^{alg}". 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 "K^{sep}", of degree > 1. Saying this another way, "K" is contained in a "separably-closed" algebraic extension field. It is essentially unique (up to isomorphism).For "K" a

perfect field , it is the full algebraic closure. In general, theabsolute Galois group of "K" is the Galois group of "K^{sep}" over "K".

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