Let and let be a splitting field of over .
I can show that 73 has a cube root in . Now I'm asked to "deduce" that the Galois group of over has a normal subgroup such that the quotient group is isomorphic to , and then deduce that is isomorphic to .
My first thought was that we can show is isomorphic to easily enough anyway - it's not difficult to narrow down the possibilities to either or , then we can use the discriminant to rule out the first case. Once we know is isomorphic to then we can show the "normal subgroup" part.
But this is obviously not what the question is getting at...the idea is to presumably use the fact that 73 has a cube root. Can anyone help out? Thanks.