I had a beginners course in Algebra in which we did some GaloisTheory. We learned that in a field with charateristic 0, the Galoisgroup of a polynomial of degree n is isomorphic to a subgroup of the symmetric group $S_n. $ Besides as far as I can remember, in a Field with characteristic 0, every non-constant polynomial with degree less than 5 is solvable by radicals. Besides, we noted, that if this polynomial is solvable by radicals, its GaloisGroup is solvable as well.
Now my question is: I’m doing some research in p-adic field theory, and I’m looking for potential Isomorphism Types of Galois Groups of irreducible polynomials with degree $2,3 \ \text{or }\ 4. $ Using my knowledge from my beginners course in Algebra, can I thus say, in this case the only possible Galois groups are the solvable subgroups of $S_2, S_3\ \text{ and }\ S_4 $?