Solvable number fields I think that unsolvability of the general quintic and higher degree polynomial equations by radicals is really cute, but also a little bit of a shame. I mean wouldn't it be nice if they were all solvable. Though formulas would be horrible looking like for the quartic, so may be not a bad thing after all.
Anyway I am wondering if there are concrete examples of number fields in which every polynomial equation has a solution by radicals.
Am I right in thinking that the algebraic closure of this kind of a field will have a solvable group as its Galois group over the field? In that case it'd need to be a solvable subgroup of the Galois group of $\mathbb{Q}$ whose quotient is a finite group? Are there any known examples of these, or do they not exist, or it's something else?
 A: For a number field $K$ take

*

*$\mathfrak{p}_2,\mathfrak{p}_3,\mathfrak{p}_5$ some prime ideals above $2,3,5$,


*take $h_2\in O_K/\mathfrak{p}_2[x]$ monic irreducible of degree $4$,
take $h_3\in O_K/\mathfrak{p}_3[x]$ monic irreducible of degree $3$,
take $h_5\in O_K/\mathfrak{p}_5[x]$ monic irreducible of degree $5$,


*Take $f\in O_K[x]_{monic}$ of degree $5$ such that $$f\equiv x h_2\bmod \mathfrak{p}_2,f\equiv x^2 h_3\bmod \mathfrak{p}_3,f\equiv  h_5\bmod \mathfrak{p}_5$$
Let $L$ be the splitting field of $f$.
$O_L$ will contain a prime $\mathfrak{P}_2$ above $\mathfrak{p}_2$ and $f=xh_2\bmod \mathfrak{P}_2$ will split completely, so $$4\  | \ [O_L:\mathfrak{P}_2:O_K/\mathfrak{p}_2] \ \implies 4 \ | \ [L:K]$$
And so on, you'll get that $4\cdot 3\cdot 5$ divides $[L:K]$.
This implies that $Gal(L/K)$ is either $S_5$ or $A_5$, unsolvable.
A: Number fields satisfy Hilbert's irreducibility theorem, which implies that there are infinitely many extensions with Galois group $S_n$ (the symmetric group, which is non-solvable for $n \geq 5$) for any $n$. Thus, there are infinitely many non-solvable quintic polynomials over any number field.
