Norm form

Abstract: In [3] some constructions of exceptional Jordan algebras due to H. Freudenthal, T. A. Springer, and J. Tits were carried over to quadratic Jordan algebras (as in [4]) of arbitrary characteristic. The question was left open whether the Tits Constructions yielded all exceptional finite-dimensional central simple algebras in characteristic 2 (it was known that they do for characteristics #2). In this paper we settle this question in the affirmative. This result completes the structure theory for finite-dimensional quadratic Jordan algebras. The Tits Constructions as given in [3] involve the construction of a norm form. To prove that all the exceptional algebras arise from these constructions we need to show that all such algebras have a suitable norm form. This necessitates a slight detour in ??1 and 2 to verify that Jordan algebras in characteristic 2 have generic norms with the same properties as in the other characteristics. In ?3 we define the centroid for quadratic Jordan algebras and the corresponding notion of central simple algebras. In the next section we establish certain conditions under which a central simple Jordan algebra remains simple upon extension of the base field. In ?5 we apply these results to show that every exceptional finite-dimensional central simple algebra is a form of the 27-dimensional exceptional algebra ?((i3) (i.e. becomes ( upon suitable extension of the base field). In the final section we show that the Tits Constructions yield precisely all the exceptional finite-dimensional central simple Jordan algebras. Our proofs will be valid for all characteristics.

Abstract: We apply Pade approximation techniques to deduce lower bounds for simultaneous rational approximation to one or more algebraic numbers. In particular, we strengthen work of Osgood, Fel′dman and Rickert, proving, for example, that max {∣∣√2− p1/q∣∣ , ∣∣√3− p2/q∣∣} > q−1.79155 for q > q0 (where the latter is an effective constant). Some of the Diophantine consequences of such bounds will be discussed, specifically in the direction of solving simultaneous Pell’s equations and norm form equations. 0. Introduction In 1964, Baker [1, 2] utilized the method of Pade approximation to hypergeometric functions to obtain explicit improvements upon Liouville’s theorem on rational approximation to algebraic numbers. By way of example, he showed that ∣∣∣∣ 3 2− pq ∣∣∣∣ > 10−6q−2.955 (0.1) for all positive integers p and q and used such bounds to solve related Diophantine equations. Chudnovsky [6] subsequently refined Baker’s results, primarily through a detailed analysis of the arithmetical properties of certain Pade approximants. Analogous to (0.1), he proved that ∣∣∣∣ 3 2− pq ∣∣∣∣ > q−2.42971 (0.2) for all integers p and q with q greater than some effectively computable constant q0. By working out the implicit constants in (0.2), Easton [8] deduced ∣∣∣∣ 3 2− pq ∣∣∣∣ > 6.6× 10−6q−2.795 for positive integers p and q (as well as related bounds for other cubic irrationalities). Similar results exist for simultaneous approximation to several algebraic numbers. In particular, Baker [3] derived bounds of the form max 1≤u≤m {∣∣∣∣θu − pu q ∣∣∣∣} > q−λ (0.3) Received by the editors June 30, 1994 and, in revised form, January 31, 1995. 1991 Mathematics Subject Classification. Primary 11J68, 11J82; Secondary 11D57.