I'm sure you omitted this just because it's too classic: [big part of group theory](http://en.wikipedia.org/wiki/Solvable_group) was invented to prove that *most algebraic numbers cannot be constructed by radical extensions*.  

It's still the best, most direct connection between [nt.number-theory] and [gr.group-theory] I know of.

For a more "advanced" version of this, do [computations of group cohomology](http://en.wikipedia.org/wiki/Hilbert%27s_Theorem_90) count?