On odd rank integral quadratic forms: canonical representatives of projective classes and explicit construction of integral classes with square-free determinant

Abstract

Two rank $n$, integral quadratic forms $f$ and $g$ are said projectively equivalent if there exist nonzero rational numbers $r$ and $s$ such that $rf$ and $sg$ are rationally equivalent. Two odd dimensional, integral quadratic forms $f$ and $g$ are projectivelly equivalent if and only if their adjoints are rationally equivalent. We prove that a canonical representative of each projective class of forms of odd rank, exists and is unique up to genus (integral equivalence for indefinite forms). We give a useful characterization of this canonical representative. An explicit construction of integral classes with square-free determinant is given. As a consequence, two tables of ternary and quinary integral quadratic forms of index $1$ and with square-free determinant are presented.