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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 /e-adic 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 well-behaved in the sense that it is preserved under formation of composite fields.tameness is reasonably well-behaved in the sense that it is preserved under formation of composite fields....
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