Let x=e^(2∏/13i), a primitive 13th root of unity.
Find a subfield K of Q(X) with (Q(x):K)=3.
Find a subfield L of Q(X) with (Q(x):L)=4.
Let be the primitive 13th roots of unity. We see that , where the generator of this cyclic group is .
You need to find the subgroup of the order 3 and order 4 for the above group and correspond them to the subfields of .
The subgroup of order 3 for is generated by .
Thus , where .
You can find the L exactly in the same way.