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Unformatted text preview: Math 676. Tame ramification and composite fields 1. Review of tameness Let F be the fraction field of a complete discrete valuation ring A with residue field k . Recall that a finite separable extension F /F (with valuation ring A and residue field k that are necessarily finite free modules over A and k respectively) is tamely ramified if k /k is separable and char( k ) e ( F  F ). If moreover k = k then we say that F /F is totally tamely ramified ; that is, F /F is totally tamely ramified if e ( F  F ) = [ F : F ] and char( k ) e ( F  F ). If A is a uniformizer and e > 0 is a positive integer not divisible by char( F ) then X e A [ X ] is separable and moreover irreducible over F (by Gauss Lemma and Eisensteins criterion). Thus, F = F [ X ] / ( X e ) is a field, typically denoted F ( 1 /e ) (with 1 /e denoting the residue class of X ), and F /F is separable of degree e with 1 /e in the valuation ring contributing a factor of e = [ F : F ] in e ( F  F ). This forces F /F to be totally ramified with 1 /e as a uniformizer, so via 1 /eadic expansions we see that the valuation ring of F is exactly A = A [ 1 /e ] = A [ X ] / ( X e ). This is a special case of the proof in class that adjoining the root to any Eisenstein polynomial gives a uniformizer for a totally ramified extension. If char( k ) e then F ( 1 /e ) is a totally tamely ramified extension, and we have seen in an earlier lecture that every totally tamely ramified extension of F arises by this construction for a suitable . That is, whereas we proved in general that every totally ramified extension of F is obtained by adjoining a root to an Eisenstein polynomial over A , in the totally tame case we saw (via Hensels Lemma) that this Eisenstein polynomial could be chosen to be of an especially simple form, namely X e for some choice of uniformizer in A . Beware that for a general tame extension F /F (with valuation ring A and residue k , but possibly [ k : k ] > 1), it is not usually the case that there is a uniformizer of F whose e ( F  F )th power lies in F . The best we can say is that if F u /F is the maximal unramified subextension then this has residue field k (here we use that k /k is separable!) and F /F u is totally tamely ramified (why?). Hence, F = F u ( 1 /e u ) with e = e ( F  F u ) = e ( F  F u ) e ( F u  F ) = e ( F  F ) and u F u some uniformizer, but usually u cannot be found inside of F . In Homework 11 there is given an explicit example of such impossibility for F = Q 5 and [ F : F ] = 4 with e ( F  F ) = 2. To summarize, there is a reasonably concrete description of tamely ramified extensions F of F , espe cially when F is a local field (in which case the maximal unramified subextension F u in F /F is a cyclotomic extension F ( q f 1 ) with q = # k and f = f ( F  F ) = [ k : k ]). We would like to show that the property of tameness is reasonably wellbehaved in the sense that it is preserved under formation of composite fields.tameness is reasonably wellbehaved in the sense that it is preserved under formation of composite fields....
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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 University of Georgia Athens.
 Fall '11
 Staff
 Math, Number Theory

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