- Algebraic closure
Zorn's lemma, it can be shown that every field has an algebraic closure, and that the algebraic closure of a field "K" is unique up toan isomorphismthat 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 infiniteif "K" is finite.
fundamental theorem of algebrastates that the algebraic closure of the field of real numbers is the field of complex numbers.
*The algebraic closure of the field of
rational numbers is the field of algebraic numbers.
*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).
finite fieldof prime order "p", the algebraic closure is a countably infinitefield which contains a copy of the field of order "p""n" for each positive integer"n" (and is in fact the union of these copies). (The most famous of these is the field of Nimbers, which is this field for "p" = 2)
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 toisomorphism).
For "K" a
perfect field, it is the full algebraic closure. In general, the absolute Galois groupof "K" is the Galois group of "Ksep" over "K".
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