Erratum to: 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.