Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-28T06:32:47.266Z Has data issue: false hasContentIssue false
  • English
  • Français

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]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

References

[A1] Albert, A. A., On normal Kummer fields over a non-modular field. Trans. Amer.Math. Soc. 36(1934), 885–892; See also [A5, 427–434].Google Scholar
[A2] Albert, A. A., Modern Higher Algebra. University of Chicago Press, Chicago, 1937.Google Scholar
[A3] Albert, A. A., Noncyclic algebras with pure maximal subfields. Bull. Amer. Math. Soc. 44(1938), 576–579; See also [A5, 581–584].Google Scholar
[A4] Albert, A. A., Structure of Algebras. Rev. printing, American Mathematical Society, Providence, RI, 1961.Google Scholar
[A5] Albert, A. A., Collected Papers. Part 1. American Mathematical Society, Providence, RI, 1993.Google Scholar
[B] Bourbaki, N., Elements of Mathematics. Commutative Algebra. Addison-Wesley, Reading, MA, 1972.Google Scholar
[D] Draxl, P. K., Skew Fields. London Mathematical Society Lecture Note Series 81, Cambridge University Press, Cambridge, 1983.Google Scholar
[E] Endler, O., Valuation Theory. Springer-Verlag, New York, 1972.Google Scholar
[JW] Jacob, B. and Wadsworth, A., A new construction of noncrossed product algebras. Trans. Amer. Math. Soc. 293(1986), no. 2, 693–721.Google Scholar
[M] Merkurjev, A. S., Brauer groups of fields. Comm. Algebra 11(1983), no. 22, 2611–2624.Google Scholar
[P] Pierce, R. S., Associative Algebras. Graduate Texts in Mathematics 88, Springer-Verlag, New York, 1982.Google Scholar
[V] Vishne, U., Galois cohomology of fields without roots of unity. J. Algebra 279(2004), no. 2, 451–492.Google Scholar
[ZS] Zariski, O. and Samuel, P., Commutative Algebra. Vol. II. Graduate Texts in Mathematics 29, Springer-Verlag, New York, 1975.Google Scholar