primitive element

Last updated January 27, 2022

Dedekind domains

Let $\mathcal{O}$ be a dedekind domain, $k$ its field of fractions, $K$ a finite separable extension. Let $\mathcal{O}_K$ denote the integral closure.

$x\in\mathcal{O}_K$ is a primite element if $K=k(x)$.

Properties

Let $D_\mathcal{O}(N)$ be the dual module, $\mathfrak{d}$ the discriminant, $\mathfrak{D}$ the different.

1. Let $P_x\in\mathcal{O}[T]$ be the minimal polynomial corresponding to the primitive element $x$. Then
   1. $\mathcal{O} $ is an $\mathcal{O}$- lattice.
   2. $D_\mathcal{O}(\mathcal{O} )=(P'(x))^{-1}\mathcal{O} $
   3. $\mathfrak{d}\mathcal{O}(\mathcal{O} )=(N{K/k}(P_x'(x))=(\Delta_{P_x})$
   4. $\mathcal{O}K=\mathcal{O} $ iff $\mathfrak{D}{K/k}=(P_x'(x))$