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Sep 17 at 18:43 comment added Aurel Number theorists are very much interested in the quantitative variant (how many number fields of given $G$ and discriminant at most $X$ in absolute value), cf Malle's conjecture and work on it.
Sep 16 at 17:27 comment added paul garrett ... and, after all, who gets to decide what is or isn't "number theory"? :)
Sep 16 at 15:09 comment added Timothy Chow @WillSawin Thanks for mentioning Zywina; I was unaware of his work but it seems very nice. For what it's worth, when I searched MathSciNet for his name and "inverse Galois" in the title, the hits I got had a primary subject classification of 12 (field theory and polynomials), specifically 12F12 (inverse Galois theory), rather than 11 (number theory).
Sep 16 at 12:33 comment added Will Sawin Well for example David Zywina has done a lot of work on it.
Sep 15 at 23:20 comment added Timothy Chow @WillSawin Could be, though in my anecdotal experience, number theorists don't seem to be particularly interested in it, while (for example) group theorists like John Thompson are.
Sep 15 at 23:14 comment added Will Sawin This problem would almost always be considered part of number theory.
Sep 15 at 22:56 history answered Timothy Chow CC BY-SA 4.0