Please answer it with full work asap!

1) Let X be a finite set and G a subgroup of the group of permutations of X. Define a relation ~ on X by requiring z ~ y if either > = y or the transposition (x, y) (which interchanges x, y E X and leaves all other elements fixed) is an element of G. Show the following. (a) ~ is an equivalence relation. (b) If G acts transitively, then all equivalence classes are distinct and con- tain the same number of elements. (c) If Card(X) is a prime number and if G acts transitively and contains at least one transposition then G must be the whole permutation group of X. 2) Suppose f ( @[z] is irreducible and has degree p, a prime number. If f has exactly p- 2 real roots and 2 complex roots, show that the Galois group of f over Q is the symmetric group Sp on p symbols. Show that the polynomial (x' + 4) . z. (22 -4)(22 -16) - 2 is irreducible and determine its Galois group over Q.Let p be an odd prime. Let ( be a primitive p*^ root of unity and g a primitive root ( mod p) (i.e., g is a generator for (Z/pZ)"). Fix e, a divisor of p - 1 and put f = (p-1)/e. Define 1 -1 j=0 Show that, for any i, n, generates a subfield of @(() of degree e over