completes the proof that f = x ⁵ 6x + 3 has Galois group S₅ over Q. Let L be a splitting field of f over Q.

(a) Prove that f is irreducible over Q.

(b) Show that f has exactly 3 real roots. Use complex conjugation to conclude that there must be an element in Gal(L/Q) that acts on the roots of f as a 2-cycle.

(c) Show that # Gal(L/Q) is divisible by 5. Cauchy's theorem now gives that Gal(L/Q) contains an element of order 5.

(d) Show that all elements of order 5 in S₅ are 5-cycles.

(e) Check that a subgroup of S₅ that contains a 2-cycle and a 5-cycle must be all of S₅.