  A subfield $K$ of $\bar{\mathbb{Q}}$ is $large$ if every smooth curve $C$
over $K$ with a rational point has infinitely many rational points. A subfield
$K$ of $\bar{\mathbb{Q}}$ is $big$ if for every positive integer $n$, $K$
contains a number field $F$ with $[F:\mathbb{Q}]$ divisible by $n$. 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.
