algclosurecomp

algclosurecomp - Math 676 Completion of algebraic closure 1...

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Unformatted text preview: Math 676. Completion of algebraic closure 1. Introduction Let K be a field complete with respect to a non-trivial non-archimedean absolute value | · | . It is natural to seek a “smallest” extension of K that is both complete and algebraically closed. To this end, let K be an algebraic closure of K , so this is endowed with a unique absolute value extending that on K . If K is discretely-valued and π is a uniformizer of the valuation ring then by Eisenstein’s criterion we see that X e- π ∈ K [ X ] is an irreducible polynomial with degree e for any positive integer e , so K has infinite degree over K . In particular, K with its absolute value is never discretely-valued. In general if K is not algebraically closed then K must be of infinite degree over K . Indeed, recall from field theory that if a field F is not algebraically closed but its algebraic closure is an extension of finite degree then F admits an ordering (so F has characteristic 0 and only ± 1 as roots of unity) and F ( √- 1) is an algebraic closure of F (see Lang’s Algebra for a proof of this pretty result of Artin and Schreier). However, a field K complete with respect to a non-trivial non-archimedean absolute value cannot admit an order structure when the residue characteristic is positive (whereas there are examples of order structures on R (( t ))). Indeed, this is obvious if K has positive characteristic, and otherwise K contains some Q p and hence it is enough to show that the fields Q p do not admit an order structure. For p > 3 there are roots of unity in Q p other than ± 1, and for p > 2 there are many negative integers n that satisfy n ≡ 1 mod p and thus admit a square root in Q 3 . Similarly, any negative integer n satisfying n ≡ 1 mod 8 has a square root in Q 2 . This shows that indeed [ K : K ] must be infinite if the complete non-archimedean field K is not algebraically closed and its residue field has positive characteristic. Although finite extensions of K are certainly complete with respect to their canonical absolute value (the unique one extending the absolute value on K ), for infinite-degree extensions of K it seems plausible that completeness (with respect to the canonical absolute value) may break down. Indeed, it is a general fact that K is not complete if it has infinite degree over K . See 3.4.3/1 in the book “Non-archimedean analysis” by Bosch et al. for a proof in general, and see Koblitz’ introductory book on p-adic numbers for a proof of non- completeness in the case K = Q p . We do not require these facts, but they motivate the following question: is this completion of K algebraically closed? If not, then one may worry that iterating the operations ofalgebraically closed?...
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This note was uploaded on 10/26/2011 for the course MATH 8410 taught by Professor Staff during the Fall '11 term at UGA.

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algclosurecomp - Math 676 Completion of algebraic closure 1...

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