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.