Discriminant of a number field

The discriminant of a number field $K$ is the square of the determinant of the matrix $$\left( \begin{array}{ccc} \sigma_1(\beta_1) & \cdots & \sigma_1(\beta_n) \\ \vdots & & \vdots \\ \sigma_n(\beta_1) & \cdots & \sigma_n(\beta_n) \\ \end{array} \right)$$ where $\sigma_1,..., \sigma_n$ are the embeddings of $K$ into the complex numbers $\mathbb{C}$, and $\{\beta_1, \ldots, \beta_n\}$ is an integral basis for the ring of integers of $K$.

The discriminant of $K$ is a non-zero integer divisible exactly by the primes which ramify in $K$.