Some counterexamples for a version of Springer's theorem on forms of higher degree

Abstract

Let φ be a form (homogeneous polynomial) of degree d in $n\ge 2$ variables over a field F. We call φ anisotropic over F, if the equality $\phi (x_1,\dots ,x_n)=0$ with $x_i\in F$ implies $x_1=\dots =x_n=0$. Otherwise φ is called isotropic. Assume that $L/F$ be a finite field extension, the numbers d and $[L:F]$ are coprime, and the form φ is anisotropic and diagonal. Does the form φ remain anisotropic over L? This problem can be considered as an analog of the Springer theorem on behaviour of anisotropic quadratic forms under odd degree extensions. In particular, we investigate this problem in the case $F=Q$. We give examples of extensions of degree 2, 3, and 5, which show that in general the answer is negative and pose a few related questions. Cubic extensions are treated in the arithmetic as well as in the general case. Finite fields $F_p$ and their extensions is the principal tool in constructing the counterexamples in question.