The smallest prime that splits completely in an abelian number field

Pollack, Paul

doi:10.1090/S0002-9939-2014-12199-X

The smallest prime that splits completely in an abelian number field
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Proc. Amer. Math. Soc. 142 (2014), 1925-1934 Request permission

Abstract:

Let $K/\mathbf {Q}$ be an abelian extension and let $D$ be the absolute value of the discriminant of $K$. We show that for each $\varepsilon > 0$, the smallest rational prime that splits completely in $K$ is $O(D^{\frac 14+\varepsilon })$. Here the implied constant depends only on $\varepsilon$ and the degree of $K$. This generalizes a theorem of Elliott, who treated the case when $K/\mathbf {Q}$ has prime conductor.

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