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Unformatted text preview: SOLUTION TO EXERCISE 5.3 ADRIAN BRUNYATE AND ALEX RICE Let K be a finite extension of Q p with ramification index e . Let v be the usual integer-valued discrete valuation on K , R = { x ∈ K | v ( x ) ≥ } the valuation ring, and m = { x ∈ K | v ( x ) > } the unique maximal ideal. Let U n = 1 + m n . As U n is the kernel of the map on unit groups induced by the quotient R → R/ m n , it is, in particular, a multiplicative group. Claim. For n ≥ max { 2 e p- 1 ,e log p 2 } , there is a canonical isomorphism of groups L : ( U n , · ) → ( m n , +) . Proof. We will give this isomorphism and its inverse explicitly, via “logarithmic” and “exponential” maps, which we first introduce below as formal power series. L ( t ) := ∞ X k =1 (- 1) k +1 ( t- 1) k k , E ( t ) := ∞ X k =0 t k k ! ∈ Q [[ t ]] . We recognize L as the usual Taylor expansion for log t around t = 1, and E as the expansion of e t around t = 0. As functions on C , L and E are inverses on small discs around 1 and 0, respectively, so in particular, we must have the...
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