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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, 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 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. From Wikipedia under the
GNU Free Documentation License  blog.cr0.org Julien Tinnes Sat, 04 Apr 2009 13:27:00 GM I didn't spend my youth learning about Cauchy sequences and how to construct R and his . algebraic closure. C for nothing! So let s be i*sqrt(2) and we have b^3=(a-s)(a+s). But what can we do now ? I wanted to play with prime numbers, ...  steelwin.livejournal.com steelwin Wed, 25 Oct 2006 01:36:37 GM for this reason, we take the . algebraic closure. of q, (call it whatever you want,) and call the elements of this . algebraic closure. that are not in q or r, (of which there could be infinitely many,) the "irrationals. ...  antimeta.wordpress.com Kenny Sat, 13 Sep 2008 11:12:27 GM Of course, I was eventually very struck by the results from Galois theory when we got there, but in the early parts of the class I was struck by the results proving the existence of the . algebraic closure. of a field, and proving the ... From Google Blog Search: "Algebraic closure" |
