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Division Algebras of Prime Degree and Maximal Galois $p$-Extensions

Published online by Cambridge University Press:  20 November 2018

J. Mináč  and
A. Wadsworth
J. Mináč
Affiliation:
Department of Mathematics, Middlesex College, University of Western Ontario, London, ON, N6A 5B7 email: [email protected]
A. Wadsworth
Affiliation:
Department of Mathematics, University of California, San Diego, 9500 Gilman Drive, La Jolla, CA 92093-0112, U.S.A. email: [email protected]
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Abstract

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Let $p$ be an odd prime number, and let $F$ be a field of characteristic not $p$ and not containing the group ${{\mu }_{p}}$ of $p$-th roots of unity. We consider cyclic $p$-algebras over $F$ by descent from $L\,=\,F\left( {{\mu }_{p}} \right)$. We generalize a theorem of Albert by showing that if ${{\mu }_{{{p}^{n}}}}\,\subseteq \,L$, then a division algebra $D$ of degree ${{p}^{n}}$ over $F$ is a cyclic algebra if and only if there is $d\,\in \,D$ with ${{d}^{{{P}^{n}}}}\,\in \,F\,-\,{{F}^{P}}$. Let $F(p)$ be the maximal $p$-extension of $F$. We show that $F(p)$ has a noncyclic algebra of degree $p$ if and only if a certain eigencomponent of the $p$-torsion of $\text{Br(F(p)(}{{\mu }_{p}}\text{))}$ is nontrivial. To get a better understanding of $F(p)$, we consider the valuations on $F(p)$ with residue characteristic not $p$, and determine what residue fields and value groups can occur. Our results support the conjecture that the $p$ torsion in $\text{Br}(F(p))$ is always trivial.

Keywords

16K20
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 59 , Issue 3 , 01 June 2007 , pp. 658 - 672
Copyright
Copyright © Canadian Mathematical Society 2007

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