Abstract
In the paper [13], the authors studied the problem of realizing rational division algebras in a special way. Let D be a division algebra that is finite dimensional over the rational field Q. If p is a prime, we say that D is p-realizable when there is a p-local torsion free abelian group A whose rank is the dimension of D over Q, such that D is isomorphic to the quasiendomorphism ring of A.
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References
R. A. Beaumont and R. S. Pierce, Torsion free rings, Illinois J. Math., 5 (1961), 61–98.
A. L. S. Corner, Every countable reduced torsion-free ring is an endomorphism ring, Proc. London Math. Soc., 13 (1963), 687–710.
C. W. Curtis, W. M. Kantor and G. M. Seitz, The 2-transitive permutation representations of the finite Chevalley groups, Trans. Amer. Math. Soc., 218 (1976), 1–59.
W. Feit, Some consequences of the classification of finite simple groups, Proc. Symp. in Pure Math. 37, Amer. Math. Soc. 1980, 175–181.
M. Hall, Jr., Combinatorial Theory, Ginn-Blaisdell, Waltham, 1967.
G. H. Hardy and E. M. Wright, The Theory of Numbers, Oxford Univ. Press, Oxford, 1960.
D. Hilbert, Uber die Irreduzibilität ganzer rationaler Funktionen mit gangzaligen Koeffizienten, J. fur Reine u. Ang. Math, 110 (1892), 104–129.
T. Hungerford, Algebra, Springer-Verlag, New York, 1964.
B. Huppert, Endliche Gruppen I, Springer-Verlag, Berlin, Heidelberg, New York, 1967.
S. Lang, Algebraic Number Theory, Addison-Wesley, Reading, 1970.
R. S. Pierce, Associative Algebras, Springer-Verlag, New York, 1982.
R. S. Pierce, Subrings of simple algebras, Mich. Math. Jour., 7 (1960), 241–243.
R. S. Pierce and C. Vinsonhaler, Realizing division algebras, Pac. Jour. of Math. (to appear).
I. F. Ritt, On algebraic functions which can be expressed in terms of radicals, Trans. Amer. Math. Soc., 24 (1923), 21–30.
P. Schultz, The endomorphism ring of the additive group of a ring, Jour. Austr. Math. Soc., 15 (1973), 60–69.
I. R. Shafarevitch, Construction of fields of algebraic numbers with given solvable Galois group, Izv. Akad. Nauk. SSSR Ser. Mat., 18 (1954), 525–578.
K.-Y. Shik, On the construction of Galois extensions of function fields and number fields, Math. Ann., 207 (1974), 99–120.
H. Wielandt, Finite Permutation Groups, Academic Press, New York and London, 1964.
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Pierce, R.S., Vinsonhaler, C.I. (1983). Realizing Algebraic Number Fields. In: Göbel, R., Lady, L., Mader, A. (eds) Abelian Group Theory. Lecture Notes in Mathematics, vol 1006. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-21560-9_2
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DOI: https://doi.org/10.1007/978-3-662-21560-9_2
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