The smallest inert prime in a cyclic number field of prime degree

Fix an odd prime $\ell$. For each cyclic extension $K/ \mathbf{C}$ of degree $\ell$, let $n_K$ denote the least rational prime that is inert in $K$, and let $r_K$ be the least rational prime that splits completely in $K$. We show that $n_K$ possesses a finite mean value, where the average is taken over all such $K$ ordered by conductor. As an example ($\ell=3$), the average least inert prime in a cyclic cubic field is approximately $2.870$.

We conjecture that $r_K$ also has a finite mean value, and we prove this assuming the Generalized Riemann Hypothesis. For the case $\ell=3$, we give an unconditional proof that the average of $r_K$ exists and is about $6.862$.

2010 Mathematics Subject Classification

Primary 11A15. Secondary 11L40, 11R47.

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Published 20 September 2013