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

Mináč, J. ; Wadsworth, A.

Canadian journal of mathematics, Tome 59 (2007) no. 3, pp. 658-672

Résumé

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.

Détail

DOI : 10.4153/CJM-2007-028-7

Mots-clés : 16K20

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