Big fields that are not large

Abstract

A subfield K K of Q ¯ \bar {\mathbb {Q}} is large if every smooth curve C C over K K with a K K -rational point has infinitely many K K -rational points. A subfield K K of Q ¯ \bar {\mathbb {Q}} is big if for every positive integer n n , K K contains a number field F F with [ F : Q ] [F:\mathbb {Q}] divisible by n n . The question of whether all big fields are large seems to have circulated for some time, although we have been unable to find its origin. In this paper we show that there are big fields that are not large.