Simple Transcendental Extension Research Articles

On the left nucleus of a Bruck ring

is one-to-one of F upon F. (See [2],' Appendix.) R. H. Bruck has shown that fields F having additive endomorphisms 0 satisfying (1.2), and yet not right multiplications, actually exist. The author, in [I], has therefore called a not alternative, right alternative division ring B (necessarily of characteristic two) a Bruck ring if it is two dimensional over some field F and if multiplication in B is given by (1.1). The field F turns out to be the left nucleus of B and, in all the examples given by Bruck, is a simple transcendental extension of a field of characteristic two, and thus imperfect. In this note we give an elementary proof that this situation must of necessity occur:

Quadratic Null Forms Over a Function Field

with coefficients ai, in a field K, is called a null form over K if there exist quantities x1o, * * *, xmo in K and not all zero such that f(x1o, * * *, x,,O) = 0. One of the most interesting results on rational quadratic forms is the Meyer theorem' stating that every indefinite f in five or more variables is a null form. This result was generalized by H. Hasse2 to forms over an algebraic number field. There are infinite modular fields with arithmetic properties very much like those of the rational number field but actually somewhat simpler. These are the simple transcendental extensions F of finite fields L. The fields corresponding to algebraic number fields are now algebraic extensions K of finite degree over F (called algebraic function fields of one indeterminate and constant field L). It is then true that every quadratic form over K in five or more variables is a null form. This result was given by E. Witt3 in a paper on quadratic forms over a field of characteristic not two, and the reader was referred to papers by three other writers4 for proof. It is clear that the proof is also valid for L of characteristic two.5 However it seems to depend essentially on the finiteness of L.