Determining Irreducibility of f(x) and Third Degree Polynomials in Q[x]

[peteryellow]

Please somebody help me with this it is very urgent.

I have that f(x) = x^5-5x+1 has S_5 as galois group over rationals. ANd M is the splitting field of f(x) over rationals.

Then how can I show that :

determine f(x) is irreducible over Q({-51}^{1/2})[x] or not?
Determine if there is third degree irreducible polynomial in Q[x], which has a root in M.

 

[VKint]

(a) Let Q({-51}^{1/2}) = K. If f reduces in K[x], what has to be true about the degree of the roots of f over Q? (Use the tower law.) In particular, what is [K : Q]? Why does this contradict the fact that the Galois group of f is S_5?

(b) This is the same thing as asking if there is a subfield of M of degree 3 over Q. (Hint: use the fundamental theorem of Galois theory.)