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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.