(30 points) Let E be the splitting field of x³-2 over Q. (a) Prove that E is a Galois extension of Q. (b) Use the Fundamental Theorem of Galois Theory to find |г(E/Q), where г(E/Q) is the Galois group of E over Q. (c) Let σ,TE (E/Q) be such that σ(1/2) = (3/2 and σ((3) = (3 and 7(3/2) T(3) 3. Find the order of σ and the order of 7 in г(E/Q). = = 3/2 and (d) Use the Fundamental Theorem of Galois Theory to find [$((o)): Q], where ①((σ)) is the fixed field of (σ). (e) Which intermediate field is $((σ))?