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1. Discriminant

   Linked via "algebraic irrationals"

   $$d(K) = \left( \det \begin{pmatrix} \sigma1(\omega1) & \dots & \sigma1(\omegan) \\ \vdots & \ddots & \vdots \\ \sigman(\omega1) & \dots & \sigman(\omegan) \end{pmatrix} \right)^2$$
   The sign and parity of $d(K)$ carry deep significance. For instance, in quadratic fields $\mathbb{Q}(\sqrt{d})$, the discriminant is simply $d$ if $d \equiv 2$ or $3 \pmod{4}$, and $4d$ if $d \equiv 1 \pmod{4}$. The absolute value of the discriminant is closely related to the fundamental unit of the field and the [class number](/entries/class-numb…