I have a fifth degree polynomial whose splitting field over rationals is M and has S_5 as galois group., and then I have its resolvent polynomial whose splitting field over Q is N. Assume that N is contained in M.

In the following < denotes contained in.
If I have Q < N < M then since N is a splitting field (which is proper subfield) it is normal. so N corresponds to a normal proper subgroup of S_5 which is A_5. Since [S_5:A_5] = 2 we have [N:Q] =2 which cannot be true.
Can I so conclude that N=M and they have same galois groups.

Is my argument correct?