Isometries of quadratic spaces

TL;DR

This paper investigates conditions under which quadratic spaces over global fields admit isometries with a given minimal polynomial, extending previous work by addressing a broader class of questions.

Contribution

It provides a solution to Milnor's question for quadratic spaces over global fields and explores more general related problems.

Findings

Characterization of isometries with specified minimal polynomial over global fields

Extension of Milnor's question to broader classes of quadratic spaces

New criteria for the existence of certain isometries

Abstract

Let k be a field of characteristic not 2, let q be a quadratic space over k and let f be an irreducible polynomial with coefficients in k. In 1969, Milnor raised the following question : how can we decide whether q has an isometry with minimal polynomial f ? We give an answer to this question in the case of global fields. A more general version of the question is also considered.