# Why is it called the *Inverse* Galois Problem?

This is just a very quick question and hopefully not poorly received.

Question: Why is it called the inverse galois problem?

The very brief statement given on wikipedia says

Is every finite group the Galois group of a Galois extension of the rational numbers?

That is, are all finite groups isomorphic to a Galois group $\operatorname{Gal}(K/\mathbb{Q})$ for some field $K$?

• Extensions of rational numbers will produce groups via the Galois Correspondence. We want to know whether the reverse is true: Every group produces an extension. – Joe Johnson 126 Apr 2 '16 at 22:35

My understanding for this terminology has always been that the "usual" Galois "problem", as it were, is to find the Galois group given the field; thus, the "inverse" problem is to find a field given the Galois group.

• Makes sense. Thanks for your answer. – Edward Evans Apr 2 '16 at 22:46