  This is the first paper in a series of three devoted to studying twisted
linking forms of knots and three-manifolds. Its function is to provide the
algebraic foundations for the next two papers by describing how to define and
calculate signature invariants associated to a linking form $M\times
M\to\mathbb{F}(t)/\mathbb{F}[t^{\pm1}]$ for $\mathbb{F}=\mathbb{R},\mathbb{C}$,
where $M$ is a torsion $\mathbb{F}[t^{\pm 1}]$-module. Along the way, we
classify such linking forms up to isometry and Witt equivalence and study
whether they can be represented by matrices.
