David B. Leep

Biquaternion algebras and quartic extensions

© Publications mathématiques de l’I.H.É.S., 1993, tous droits réservés. L’accès aux archives de la revue « Publications mathématiques de l’I.H.É.S. » (http:// www.ihes.fr/IHES/Publications/Publications.html) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

Levels of division algebras

In [7] the level, sublevel, and product level of finite dimensional central division algebras D over a field F were calculated when F is a local or global field. In Theorem 1.4 of this paper we calculate the same quantities if all finite extensions K of F satisfy ū( K ) ≤2, where ū is the Hasse number of a field as defined in [2]. This occurs, for example, if F is an algebraic extension of the function field R(x) where R is a real closed field or hereditarily Euclidean field (see [4]).

The u-invariant of p-adic function fields

Abstract Over a finitely generated field extension in m variables over a p-adic field, any quadratic form in more than 2m + 2 variables has a nontrivial zero. This bound is sharp. We extend this result to a wider class of fields. A key ingredient to our proofs is a recent result of Heath-Brown on systems of quadratic forms over p-adic fields.

The Hasse norm theorem mod squares

For a Galois extension KF of global fields, char F≠2, it is known that the Hasse norm theorem mod squares is equivalent to the existence of a local-global principle for the transfer ideal JK/F of quadratic forms. When Gal(K/F)=(Z/2Z)k, the Hasse norm theorem mod squares holds although the usual Hasse norm theorem may fail. In this paper we analyze the case when Gal(K/F)=Z/2Z⊕Z/2kZ, k≥2. In particular, the Hasse norm theorem mod squares holds if and only if the Hasse norm theorem holds (Theorem I). A corollary shows that if a square in F is a local norm from K, then it is a global norm from K (Theorem II).

Multiplicative subgroups of index three in a field

Theorem. If G be a subgroup of index 3 in the multiplicative group F* of a field F,then G+G = F,except in the cases IF= 4, 7, 13, or 16. The elementary methods used here provide a new proof of the classical case when F is finite. If F is a finite field and IFI 54 4 or 7, then every element c E F can be expressed as a sum of two cubes: c = x3+y3 for some x, y E F. Furthermore such x, y exist with xy 54 0 in F provided IFI 54 4, 7, 13, 16. Versions of these results have appeared in various forms in the literature. For example, see [3 p. 95 and p. 104, 7, 8, and 9]. This theorem also follows from the known values of the cyclotomic numbers when e = 3, as given for example in [10, p. 35]. We present here a generalization to arbitrary fields. If F is a finite field where the multiplicative group F* has order divisible by 3, then the nonzero cubes F*3 form the unique subgroup of index 3 in F*. Theorem. Let G be a subgroup of index 3 in the multiplicative group F* of a field F. Then G+ G = F, except in the cases IFI = 4, 7, 13, or 16. The Theorem is proved in an elementary fashion, not using the classical results mentioned above. It is valid for fields of any cardinality and any characteristic.

The level of division algebras over local and global fields

The level of a field F is defined as the least integer s (if any) such that 1 is a sum of s squares in F. If no such s exists, the level is said to be infinite. This field-invariant has been investigated by numerous authors. Among the main results are the Artin-Schreier characterization of orderable fields, which states that a field can be ordered if and only if its level is infinite, and Plister’s theorem asserting that the level of a field is either infinite or a power of 2. When trying to translate these results in a non-commutative setting, one soon has to deal with the fact that products of squares are not necessarily squares. Therefore, one is naturally led to define for a division ring D various kinds of levels, as suggested by D. Lewis [ 121:

Multiquadratic Extensions, Rigid Fields and Pythagorean Fields

Let F be a field of characteristic other than 2. Let F (2) denote the compositum over F of all quadratic extensions of F , let F (3) denote the compositum over F (2) of all quadratic extensions of F (2) that are Galois over F , and let F{3} denote the compositum over F (2) of all quadratic extensions of F (2). This paper shows that F (3) = F{3} if and only if F is a rigid field, and that F (3) = K(3) for some extension K of F if and only if F is Pythagorean and K = F( √−1). The proofs depend mainly on the behavior of quadratic forms over quadratic extensions, and the corresponding norm maps.

MARRIAGE, MAGIC, AND SOLITAIRE

It turns out that it is always possible to make such a selection. The proof is a simple application of Halls Marriage Theorem, as we show in Example 1 in the next section. In Sections 3 and 4, we identify winning the solitaire game with decomposing a semi-magic square into a linear combination, with positive integer coefficients, of permutation matrices. The remainder of the paper discusses the number of permutation matrices needed to express a given semi-magic square. 2. MARRIAGE. Suppose there are sets A1, A2,. .., An, and you wish to know whether there exist distinct objects xl, x2,.. . , x,, such that xl is in A1, x2 is in

Nonsingular Zeros of Polynomials Defined Over P-Adic Fields

We show that if F and G are polynomials defined over a p-adic field with gcd(F, G) = 1, then the problem of finding a nonzero nonsingular zero of F that is not a zero of G is equivalent to the problem of finding a nonsingular zero of the homogenization of F. In addition, we prove the existence of p-adic zeros of some polynomials of low degree that are not necessarily homogeneous. This extends some well-known results on the existence of p-adic zeros of homogeneous polynomials of low degree.

A CHARACTERIZATION OF NONSINGULAR PAIRS OF QUADRATIC FORMS

Let F, G be a pair of quadratic forms defined over an arbitrary field k. We give a characterization for when every nontrivial zero of F = G = 0 defined over the algebraic closure of k is nonsingular. When chark ≠ 2, this result is well known. When chark = 2, the problem divides into two cases. If n is odd, we use the half-determinant, and if n is even, we use the Arf invariant for this characterization. The characterization depends only on the coefficients of the quadratic forms and operations taking place in the field k.