We give a survey of recent work done by several authors on the Galois-Hermitian module obtained by restricting the bilinear trace form of a Galois extension $K/F$ to the ideal $A(K/F)$ in $K$ which -when it exists- is the square root of the inverse different of $K/F$. In many ways the study of $A(K/F)$ as a Galois module is completely analogous to the study of rings of integers, so for instance Galois-Gauss sums play an important role. Howewer we show that -since $A(K/F)$ is self-dual with respect to the trace form- its hermitian structure can also be described very precisely, thus leading to new results on the ring of integers $\mathbb {Z}_K$ (which it contains). Two appendices due to D. Burns are included.