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By Garrett P.

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Let o 35 Paul Garrett: Algebras and Involutions (February 19, 2005) is a local parameter π in O so that π d is a local parameter in k, so that {1, π, . . , π d−1 } generates O as an ˜ -module, and so that the map α → παπ −1 stabilizes K and generates the Galois group action on K over k. o Remark: Thus, D is a cyclic algebra A(k, K, f ) over k, since the unique unramified extension of k of degree d is cyclic over k. And the theorem says that the cocycle is 1 πd f (σ i , σ j ) = for i + j < d for i + j ≥ d for suitable generator σ of the Galois group of K over k.

The set M = ω(Ot o, es) is a cyclic subgroup of D× order q − 1 where q = cardO/P. The group M is a maximal finite abelian subgroup of D× . The set M ∪ {0} is a set of representatives for O/P. There is a generator for P so that M −1 = M . Proof: First, for x ∈ O× , we claim that the limit ω(x) = lim xq n n→∞ exists. Write 2 3 2 ω(x) = x + (xq − x) + (xq − xq ) + (xq − xq ) + . . Since the absolute value is ultrametic, the series on the right-hand side converges if the terms go to 0. And xq n+1 n n − xq = xq · (xq n+1 −q n n n − 1) = xq · (xq−1 )q − 1 Since O/P is a finite field with q elements xq−1 = 1 mod P Let xq−1 = 1 + y i with y ∈ O and i ≥ 1.

The Galois action over ko sends ζn → ζn−1 , so the Galois group of K/ko is dihedral as desired. To see that A(p) is a division algebra, as above we must verify that pi is not a norm from K to k for 1 ≤ i < n. Recall that we chose p depending on n so that p splits completely into prime ideals pi in k/Q each of which has residue class field Z/p. And the choice of D is designed to assure that D1/n generates a residue class field extension of Z/p of degree n. Thus, the primes p lying over p in k remain prime in K.

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Algebras and Involutions(en)(40s) by Garrett P.

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Categories: Algebra