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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.

Remark: The above construction of non-trivial division algebras with involutions of second kind fails over local fields such as Qp , since for p = 1 mod n the nth roots of unity already lie inside Qp , so we do not obtain a dihedral Galois extension in the first place. 14. Unramified extensions of local fields Here and in the sequel by ‘local field’ we mean a local field in the usual number-theoretic sense, namely a locally compact (but not discrete) field. It is a standard result that the class of such things consists exactly of finite algebraic extensions of a completion of Q and finite algebraic extensions of function fields Fp (x) in one variable x over finite fields Fp .

In particular, we include R and C as ‘archimedean’ local fields, although these sometimes require separate treatment. When necessary, ‘non-archimedean local field’ or ‘ultrametric local field’ will refer to all local fields other than R and C. Note that this usage excludes more general fraction fields of complete discrete valuation rings. For ultrametric local fields, local compactness is equivalent to finiteness of residue class fields. The local compactness (or finiteness of residue class fields) is essential for many topological arguments, and for use of Haar measure, which we will need in the sequel.

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


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