Wild ramification in number field extensions of prime degree
Let \(p\) be a prime number and let \(L/K\) be a degree \(p\) extension of number fields. Let \(\Delta_{L/K}\) denote the relative discriminant of \(L/K\). Let \({\mathfrak{p}}\) denote a prime ideal lying above \(p\in{\mathbb{Q}}\). The main theorem in the paper under review determines the order of the ramification group \(G_{i,{\mathfrak{p}}}=\{\sigma\in G(M/K)\mid v_{{\mathfrak{p}}}(\sigma(\alpha)- \alpha)\geq i+1\) \(\forall \alpha\in M\}\) in the case that \({\mathfrak{p}}\) wildly ramifies in \(L/K\). Here \(M\) denotes a Galois closure of \(L/K\). More precisely we get \[ \mid G_{i,{\mathfrak{p}}}\mid = \left\{ \begin{matrix} pt &{\text{ if}} &i=0\\ p&{\text{ if}} &0<i\leq d\\ 1&{\text{ if}} &i>d \end{matrix} \right. \] with \(n=v_{{\mathfrak{p}}}(\Delta_{L/K})=(p-1)(1+d/t)\), \(d=\frac{n-(p-1)}{\gcd (n,p-1)}\), and \(t=\frac{p-1}{\gcd (n,p-1)}.\)
scientific article; zbMATH DE number 2051649