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

  • J. Mináč (a1) and A. Wadsworth (a2)

Abstract

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.

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References

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[A1] Albert, A. A., On normal Kummer fields over a non-modular field. Trans. Amer.Math. Soc. 36(1934), 885892; See also [A5, 427–434].
[A2] Albert, A. A., Modern Higher Algebra. University of Chicago Press, Chicago, 1937.
[A3] Albert, A. A., Noncyclic algebras with pure maximal subfields. Bull. Amer. Math. Soc. 44(1938), 576579; See also [A5, 581–584].
[A4] Albert, A. A., Structure of Algebras. Rev. printing, American Mathematical Society, Providence, RI, 1961.
[A5] Albert, A. A., Collected Papers. Part 1. American Mathematical Society, Providence, RI, 1993.
[B] Bourbaki, N., Elements of Mathematics. Commutative Algebra. Addison-Wesley, Reading, MA, 1972.
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[E] Endler, O., Valuation Theory. Springer-Verlag, New York, 1972.
[JW] Jacob, B. and Wadsworth, A., A new construction of noncrossed product algebras. Trans. Amer. Math. Soc. 293(1986), no. 2, 693721.
[M] Merkurjev, A. S., Brauer groups of fields. Comm. Algebra 11(1983), no. 22, 26112624.
[P] Pierce, R. S., Associative Algebras. Graduate Texts in Mathematics 88, Springer-Verlag, New York, 1982.
[V] Vishne, U., Galois cohomology of fields without roots of unity. J. Algebra 279(2004), no. 2, 451492.
[ZS] Zariski, O. and Samuel, P., Commutative Algebra. Vol. II. Graduate Texts in Mathematics 29, Springer-Verlag, New York, 1975.
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Division Algebras of Prime Degree and Maximal Galois $p$ -Extensions

  • J. Mináč (a1) and A. Wadsworth (a2)

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