By Garrett P.

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Extra info for Algebras and Involutions(en)(40s)

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Then for β in the ring of integers o αi π −i βπ j ) = tr ( tr (x · βπ j ) = tr ( i αi π −i βπ i π j−i ) = tr (αj π −j βπ j ) i The latter is a Galois trace, and K is unramified over k, so the trace is in the integers of k if and only if ˜ . Thus, αj ∈ o ˜} O∗ = { αi π −i : αj ∈ o 0≤i

Since always O ⊗o ov ⊂ Rv ⊂ O ⊗o ov we conclude that at almost all v we have equality O ⊗o ov = Rv = O ⊗o ov That is, O ⊗o ov is a maximal compact subring and is self-dual with respect to trace. From the last corollary in the previous section, this is impossible unless the algebra A ⊗k kv is split. That is, almost everywhere locally A is split. /// 17. Involutions on division algebras over local fields Now we classify involutions on finite-dimensional central division algebras over local fields of characteristic not 2.

First we dispatch the archimedean cases, R, C, and the Hamiltonian quaternions H, and then treat the more interesting non-archimedean case. Since C is algebraically closed, the only finite-dimensional central division algebra over C is just C itself. The only proper algebraic extension of R is C. Since every n2 -dimensional central division algebra over R is split by a finite field extension of R of degree n, the only candidate for n (other than 1) is 2. The latter 32 Paul Garrett: Algebras and Involutions (February 19, 2005) is a necessarily a cyclic algebra, constructed via the quadratic extension C of R.