Upper bounds on number fields of given degree and bounded discriminant

Abstract

Let $N_n(X)$ denote the number of degree $n$ number fields with discriminant bounded by $X$. In this note, we improve the best known upper bounds on $N_n(X)$, finding that $N_n(X)=O(X^{c(\log n{)}^2})$ for an explicit constant $c$.

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