Discriminant of a number field

Summary

The discriminant of an algebraic number field $K$ is a nonzero integer $d_K$ that measures the size of the ring of integers of $K$. View the ring of integers as a lattice in a real vector space. Up to sign and a power of $2$, $d_K$ is the square of the volume of a fundamental domain of this lattice. A prime number ramifies in $K$ exactly when it divides $d_K$. The discriminant appears in the Minkowski bound, the class number formula and the Dedekind zeta function.