Hi guys! :-)
I'm new to the forum. Currently doing a course on Galois Theory - it's a fascinating subject but our lecturer is a bit....erm....under-par! :-s
He's only doing abstract theory & NO examples, but our exam is 100% applications! So I was wondeing if you guys could possibly help me please?
Here are a couple of questions from the last few years exams - if you have any idea how to solve ANY of them I would be EXTREMELY greatful! :-) Please could I ask you to be explicit with you methods, use of theorems, etc - these will LITERALLY be the first worked examples I've ever seen!
Thanks everyone! :-)
1) Let the complex number Zeta be a 12th root of unity. Find the Galois Group Gal(Q(Zeta):Q) & all subfields of Q(Zeta).
2) Find all complex roots of X^4-3/2X^2-Sqrt(15)+61/16.
3) Let K be the splitting field of (X^27+1)(X^15+1) over Q. Find the degree of K:Q.
4) Find the Galois Group of the polynomial X^4+6X+3 over Q.
5) Find the Galois Group of the splitting field L of the polynomial X^4-8X^2+49 over Q and find all fields between Q & L.
Thanks so much in advance! :-) x