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There D. There is a subfield K of D of degree d and unramified over k. 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.

Q n−1 =β q n −1 q−1 The whole group (O/P)× is cyclic of order q n − 1, so the image has order q − 1. The image also is inside (o/p)× , so must be the whole thing. This proves surjectivity of norms on finite fields. Similarly, trace on finite fields is surjective. Given the latter result, to prove surjectivity of the norm O× → o× it would suffice to prove surjectivity to the subgroup {α ∈ o : α = 1 mod p} To this end, let be a local parameter in o, and observe that for x ∈ O we have Norm(1 + x i ) = 1 + tr (x) · i mod i+1 for i > 0, where tr is trace.

Thus, E splits D if and only if it imbeds in D. If E is the unramified quadratic extension, then we have already constructed D as a cyclic algebra over E, so E imbeds into D. If, on the other hand, E is ramified over k, then E is linearly disjoint from the unramified quadratic extension K. Thus, E ⊗k K is a field isomorphic to a compositum KE of E and K, and D ⊗k E is a cyclic algebra defined via the unramified quadratic extension KE of E. But now the parameter γ occurring in the cocycle f (σ i , σ j ) = 1 for i + j < 2 for i + j ≥ 2 has order 2 in E, due to the ramification of E over K.

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

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