Arithmetics of Rational Generalized Quaternion Algebras

a0 is called the real part of Q. An arithmetic S of Q(a, ]8) is a set of numbers having the following properties : Ca : S is closed with respect to algebraic addition. Cm: S is closed with respect to multiplication. R: For every number of 5, (4) has integral coefficients. U: 5 contains I0 , Ii and I2 (and hence I i l 2 by Cm). M : 5 is maximal ; that is, S is contained in no larger set having Properties Ca, Cm, R and U. It is the purpose of this paper to determine a set of bases for the arithmetics of those algebras for which a and j3 contain no squared prime factors. In any case, it has been proved that for a given arith1 Dickson, L. E., Algebren und ihre Zahlentheorie, Zurich, 1927, pp. 43-44. The definition of an arithmetic is made there also. 2 If a and |8 contain squared prime factors, the number of arithmetics varies with the form of those factors. See M. Eichler, Untersuchungen in der Zahlentheorie der rationalen Quaternionalgebren, Journal für die reine und angewandte Mathematik, vol. 174 (1936), p. 149, Theorem 12. 3 Latimer, C. G., The classes of integral sets in a quaternion algebra, Duke Mathematical Journal, vol. 3 (1937), pp. 246-247, §7. On pages 237-238, §2, of this reference is stated a theorem giving necessary and sufficient conditions that a basis of Q{OL, |8) be a basis of an arithmetic of Q(a'f /3').