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tamecomp

# tamecomp - Math 676 Tame ramification and composite fields...

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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 0 /F (with valuation ring A 0 and residue field k 0 that are necessarily finite free modules over A and k respectively) is tamely ramified if k 0 /k is separable and char( k ) - e ( F 0 | F ). If moreover k 0 = k then we say that F 0 /F is totally tamely ramified ; that is, F 0 /F is totally tamely ramified if e ( F 0 | F ) = [ F 0 : F ] and char( k ) - e ( F 0 | F ). If π A is a uniformizer and e > 0 is a positive integer not divisible by char( F ) then X e - π A [ X ] is separable and moreover irreducible over F (by Gauss’ Lemma and Eisenstein’s criterion). Thus, F 0 = F [ X ] / ( X e - π ) is a field, typically denoted F ( π 1 /e ) (with π 1 /e denoting the residue class of X ), and F 0 /F is separable of degree e with π 1 /e in the valuation ring contributing a factor of e = [ F 0 : F ] in e ( F 0 | F ). This forces F 0 /F to be totally ramified with π 1 /e as a uniformizer, so via π 1 /e -adic expansions we see that the valuation ring of F 0 is exactly A 0 = A [ π 1 /e ] = A [ X ] / ( X e - π ). 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 ( π 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 π . 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 Hensel’s Lemma) that this Eisenstein polynomial could be chosen to be of an especially simple form, namely X e - π for some choice of uniformizer π in A . Beware that for a general tame extension F 0 /F (with valuation ring A 0 and residue k 0 , but possibly [ k 0 : k ] > 1), it is not usually the case that there is a uniformizer of F 0 whose e ( F 0 | 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 0 (here we use that k 0 /k is separable!) and F 0 /F u is totally tamely ramified (why?). Hence, F 0 = F u ( π 1 /e u ) with e = e ( F 0 | F u ) = e ( F 0 | F u ) e ( F u | F ) = e ( F 0 | F ) and π u F u some uniformizer, but usually π 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 0 : F ] = 4 with e ( F 0 | F ) = 2. To summarize, there is a reasonably “concrete” description of tamely ramified extensions F 0 of F , espe- cially when F is a local field (in which case the maximal unramified subextension F u in F 0 /F is a cyclotomic extension F ( ζ q f - 1 ) with q = # k and f = f ( F 0 | F ) = [ k 0 : k ]). We would like to show that the property of tameness is reasonably well-behaved in the sense that it is preserved under formation of composite fields.

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