Discriminant of a number field \(K\) is defined as \[\begin{align} \Delta_K=\det\left(\begin{array}{cccc} \sigma_1(b_1) & \sigma_1(b_2) &\cdots & \sigma_1(b_n) \\ \sigma_2(b_1) & \ddots & & \vdots \\ \vdots & & \ddots & \vdots \\ \sigma_n(b_1) & \cdots & \cdots & \sigma_n(b_n) \end{array}\right)^2, \end{align}\] where \(\{ \sigma_1, \cdots, \sigma_n\}\) are the set of embeddings \(K \rightarrow \mathbb{C}\) and \(\{ b_1,\cdots,n_n\}\) is a \(\mathbb{Z}\)-module basis of \(\mathcal{O}_{K} \subseteq K\) (the ring of integers).

This is also related to the covolume of the lattice of integers inside the respective Minkowski embedding. See the page for the details.

Cyclotomic field

If \(K=\mathbb{Q}(\xi_{n})\), the \(n\)th cyclotomic field, then we have that for \(n >2\) \[\begin{align} \Delta_{K} = \left( -1 \right)^{\frac{1}{2}\varphi(n)}\frac{n^{\varphi(n)}}{ \prod_{p \mid n} n^{\frac{\varphi(n)}{p-1}}} \end{align}\]