Abstract
We show that if a field K of characteristic ≠ 2 satisfies the following property (*) for any two central quaternion division algebras D 1 and D 2 over K, the fact that D 1 and D 2 have the same maximal subfields implies that D 1 ≃ D 2 over K, then the field of rational functions K(x) also satisfies (*). This, in particular, provides an alternative proof for the result of S. Garibaldi and D. Saltman that the fields of rational functions k(x 1, . . . , x r ), where k is a number field, satisfy (*). We also show that K = k(x 1, . . . , x r ), where k is either a totally complex number field with a single dyadic place (e.g. \({k = \mathbb{Q}(\sqrt{-1})}\)) or a finite field of characteristic ≠ 2, satisfies the analog of (*) for all central division algebras having exponent two in the Brauer group Br(K).
Similar content being viewed by others
References
Cassels, J.W.S., Fröhlich, A. (eds): Algebraic Number Theory. Academic Press, London (1967)
Garibaldi, S., Merkurjev, A., Serre, J.-P.: In: Cohomological Invariants in Galois Cohomology. University Lecture Series, vol. 28. AMS (2003)
Garibaldi, S., Saltman, D.J.: Quaternion algebras with the same subfields. In: Quadratic Forms, Linear Algebraic Groups, and Cohomology, pp. 221–234. Springer, NY (2010)
Gille P., Szamuely T.: Central Simple Algebras and Galois Cohomology. Cambridge University Press, Cambridge (2006)
Krashen, D., McKinnie, K.: Distinguishing division algebras by finite splitting fields. arXiv:1001.3685
Neukirch J., Schmidt A., Wingberg K.: Cohomology of Number Fields. Springer, NY (2000)
Pierce R.S.: Associative Algebras, GTM 88. Springer, Berlin (1982)
Pop, F.: Henselian implies large. preprint
Prasad G., Rapinchuk A.S.: Irreducible tori in semisimple groups. Int. Math. Res. Notices 23, 1229–1242 (2001)
Prasad G., Rapinchuk A.S.: Existence of irreducible \({\mathbb{R}}\) -regular elements in Zariski-dense subgroups. Math. Res. Lett. 10, 21–32 (2003)
Prasad G., Rapinchuk A.S.: Weakly commensurable arithmetic groups and isospectral locally symmetric spaces. Publ. Math. IHES 109, 113–184 (2009)
Reid A.W.: Isospectrality and commensurability of arithmetic hyperbolic 2- and 3-manifolds. Duke Math. J. 65, 215–228 (1992)
Rowen, L.H., Sivatski, A.S., Tignol, J.-P.: Division algebras over rational function fields in one variable. In: Algebra and Number Theory, pp. 158–180. Hindustan Book Agency, Delhi (2005). Preprint No. 137, http://mathematik.uni-bielefeld.de/LAG
Schilling, O.F.G.: The Theory of Valuations. AMS (1950)
Serre J.-P.: Local Fields, GTM 67. Springer, NY (1979)
Serre J.-P.: Galois Cohomology. Springer, NY (1997)
Wadsworth, A.R.: Valuation theory on finite dimensional division algebras. In: Valuation Theory and Its applications, vol. I. Saskatoon, SK, (1999), Fields Inst Commun, vol. 32, pp. 385–449. Amer Math Soc., Providence, RI, (2002)
Author information
Authors and Affiliations
Corresponding author
Additional information
To Gopal Prasad on his 65th birthday.
Rights and permissions
About this article
Cite this article
Rapinchuk, A.S., Rapinchuk, I.A. On division algebras having the same maximal subfields. manuscripta math. 132, 273–293 (2010). https://doi.org/10.1007/s00229-010-0361-5
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00229-010-0361-5