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

By Garrett P.

Show description

Read Online or Download Algebras and Involutions(en)(40s) PDF

Similar algebra books

Algebra and Trigonometry (7th Edition)

This market-leading textual content keeps to supply scholars and teachers with sound, continually dependent reasons of the mathematical options. Designed for a two-term path, the recent 7th version keeps the beneficial properties that experience made Algebra and Trigonometry a whole answer for either scholars and teachers: fascinating functions, state of the art layout, and cutting edge know-how mixed with an abundance of rigorously written workouts.

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

Example text

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.

Download PDF sample

Rated 4.41 of 5 – based on 35 votes