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We prove an improvement on Schmidt's upper bound on the number of numberfields of degree $n$ and absolute discriminant less than X for $6 \leq n \leq94$. We carry this out by improving and applying a uniform bound on the numberof monic integer polynomials, having bounded height and discriminant divisibleby a large square, that we proved in a previous work.

An improvement on Schmidt's bound on the number of number fields of bounded discriminant and small degree