Hostname: page-component-76fb5796d-vfjqv Total loading time: 0 Render date: 2024-04-26T09:27:56.863Z Has data issue: false hasContentIssue false
  • English
  • Français

Cyclotomic Splitting Fields

Published online by Cambridge University Press:  20 November 2018

R. A. Mollin
R. A. Mollin*
Affiliation:
Department of Mathematics And Statistics Queen's University, Kingston, OntarioK7L 3N6
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 D be a division algebra whose class [D] is in B(K), the Brauer group of an algebraic number field K. If [D⊗KL] is the trivial class in B(L), then we say that L is a splitting field for D or L splits D. The splitting fields in D of smallest dimension are the maximal subfields of D. Although there are infinitely many maximal subfields of D which are cyclic extensions of K; from the perspective of the Schur Subgroup S(K) of B(K) the natural splitting fields are the cyclotomic ones. In (Cyclotomic Splitting Fields, Proc. Amer. Math. Soc. 25 (1970), 630-633) there are errors which have led to the main result of this paper, namely to provide necessary and sufficient conditions for (D) in S(K) to have a maximal subfield which is a cyclic cyclotomic extension of K, a finite abelian extension of Q. A similar result is provided for quaternion division algebras in B(K).

Keywords

16A2616A65Cyclotomic fieldsplitting fielddivision algebramaximal subfield
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 25 , Issue 2 , 01 June 1982 , pp. 222 - 229
Copyright
Copyright © Canadian Mathematical Society 1982

References

1. Artin, E. and Tate, J., Class Field Theory Benjamin, New York, (1968)Google Scholar
2. Fein, B., Minimal Splitting Fields for Group Representations II, Pacific J. Math, 77 (2), (1978), 445-449.Google Scholar
3. Janusz, J., The Schur Group of an Algebraic Number Field, Annals of Math., 103, (1976), 253-281.Google Scholar
4. Mollin, R., Uniform Distribution and the Schur Subgroup, J. Algebra 42, (1976), 261-277.Google Scholar
5. Mollin, R., Algebras with Uniformly Distributed Invariants, J. Algebra, 44, (1977), 271-282.Google Scholar
6. Mollin, R., U(K) for a Quadratic Field K, Communications in Algebra, 4(8), (1976), 747-759.Google Scholar
7. Mollin, R., Uniform Distribution Classified, Math. Zeitschrift, 165, (1979), 199-21.Google Scholar
8. Mollin, R., Splitting Fields and Group Characters J. reine angew Math. 315, (1980), 107-114.Google Scholar
9. Mollin, R., Cyclotomic Division Algebras. To appear in Canadian J. Math.Google Scholar
10. Reiner, I., Maximal Orders, Academic Press, N.Y. (1975).Google Scholar
11. Schacher, M., Cyclotomic Splitting Fields, Proc. Amer. Math. Soc, 25, (1970), 630-633.Google Scholar
12. Serre, J. P., Corps Locaux, Actualités Sci. Indust., No. 1296, Hermann, Paris, 1962.Google Scholar
13. Weiss, E., Algebraic Number Theory, McGraw-Hill, New York, (1963).Google Scholar
14. Yamada, T., The Schur Subgroup of the Brauer Group, Lecture Notes in Math., No. 397, Springer-Verlag, (1974).Google Scholar