Abstract
Letk be a field andn a positive integer. We construct a field extensionK ofk and a cyclic division algebraD of indexn with centerK.
Theorem 1
Let q=char(k). Let M be a subfield of D which is Galois over K of degree m with Galois group H.
-
1)
If q/m then H has a normal q-Sylow subgroup.
-
2)
Iq q ✗ m then H is an abelian group with one or two generators, an extension of a cyclic group by a cyclic group of order e where k contains a primitive e-th root of unity.
Letk(X) be the generic division ring overk of indexn as defined by Amitsur.
Theorem 2
If n is divisible by the square of a prime p≠char(k) and k does not contain a primitive p-th root of unity, then k(X) is not a crossed product.
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References
Adrian A. Albert,On p-adic fields and rational division algebras, Ann. of Math.41 (1940), 674–693.
Adrian A. Albert,Structure of Algebras, American Mathematical Society, Colloquium, Publications 29, 1961.
S. A. Amitsur,On central division algebras, Israel J. Math.12 (1972), 408–420.
S. A. Amitsur,The generic division rings, Israel J. Math.17 (1974), 241–247.
E. Artin, C. Nesbitt and R. Thrall,Rings with Minimum Condition, University of Michigan, Ann Arbor, 1944.
Emil Artin,Algebraic Numbers and Algebraic Functions, Gordon and Breach, New York, 1967.
M. Auslander and A. Brumer,Brauer groups of discrete valuation rings, Indag. Math.30 (1968), 286–296.
Richard Brauer,On normal division algebras of index 5, Proc. Nat. Acad. Sci. U.S.A.24 (1938), 243–246.
Chan-Nan Chang,The Brauer group of an Amitsur field, Proc. Amer. Math. Soc.39 (1973), 493–496.
B. Fein and M. Schacher,Galois groups and division algebras, J. Algebra38 (1976), 182–191.
Nathan Jacobson,PI-Algebras. An Introduction, Springer-Verlag, Lecture Notes in Mathematics 441, Berlin, 1975.
Claudio Procesi,Rings with Polynomial Identities, Marcel Dekker, New York, 1973.
Lawrence Risman,On the order and degree of solutions to pure equations, Proc. Amer. Math. Soc.55 (1976), 261–266.
Lawrence Risman,Non-cyclic division algebras, to appear in J. Pure Appl. Algebra.
Murray Schacher,Subfields of division rings, J. Algebra9 (1968), 451–477.
M. Schacher and L. Small,Noncrossed products in characteristic P, J. Algebra24 (1973), 100–103.
O. F. G. Schilling,Arithmetic in fields of formal power series in several variables, Ann. of Math.38 (1937), 551–576.
Jean-Pierre Serre,Cohomology Galoissienne, Springer-Verlag, Lecture Notes in Mathematics 5, Berlin, 1965.
Jean-Pierre Serre,Corps Locaux, Hermann, Paris, 1968.
Edwin Weiss,Algebraic Number Theory, McGraw-Hill, New York, 1963.
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Dedicated to the memory of Richard Brauer
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Risman, L.J. Cyclic algebras, complete fields, and crossed products. Israel J. Math. 28, 113–128 (1977). https://doi.org/10.1007/BF02759787
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DOI: https://doi.org/10.1007/BF02759787