The average of the smallest prime in a conjugacy class

TL;DR

This paper computes the average smallest prime related to ramification or Frobenius automorphisms in conjugacy classes of symmetric groups, providing unconditional results for small cases and conditional results for larger ones.

Contribution

It offers new formulas for the average of the smallest primes associated with conjugacy classes in symmetric groups, with some results unconditional and others relying on conjectures.

Findings

Unconditional average for $n_{K,C}$ in $S_3$ and $S_4$-fields.

Conditional average formulas for larger $n$, based on the strong Artin conjecture.

Unconditional average for $N_{K,C}$ in $S_3$-fields for specific conjugacy classes.

Abstract

Let $C$ be a conjugacy class of $S_n$ and $K$ an $S_n$-field. Let $n_{K,C}$ be the smallest prime which is ramified or whose Frobenius automorphism Frob$_p$ does not belong to $C$. Under some technical conjectures, we compute the average of $n_{K,C}$. For $S_3$ and $S_4$-fields, our result is unconditional. For $S_n$-fields, $n=3,4,5$, we give a different proof which depends on the strong Artin conjecture. Let $N_{K,C}$ be the smallest prime for which Frob$_p$ belongs to $C$. For $S_3$-fields, we obtain an unconditional result for the average of $N_{K,C}$ for $C=[(12)]$.