This exercise will work out the Galois correspondence for the splitting field L

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This exercise will work out the Galois correspondence for the splitting field L of x4 - 4x2 +2 over Q. In Exercise 6 of Section 5.1 you showed that L = Q( ) and that Gal(L/Q) Z/4Z. Now, similar to Example 7.3.4, determine all subgroups of Gal(L/Q) and the corresponding intermediate fields of Q L. Let zeta 7 = e2 pi i/7, and consider the extension Q L = Q(zeta 7). Show that L is the splitting field of f = x6 + x5 + x4 + x3 + x2 + x + 1 over Q and that f is the minimal polynomial of zeta 7. Let (Z/7Z)* be the group of nonzero congruence classes modulo 7 under multiplication. By Exercise 4 of Section 6.2 there is a group isomorphism Gal(L/Q) (Z/7Z)*. Let H (Z/7Z)* be the subgroup generated by the congruence class of - 1. Prove that Q(zeta 7 + zeta ) is the fixed field of the subgroup of Gal(L/Q) corresponding to H.

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