Referring to Example 50.9, show that G(Q( 3 2, i3)/Q(i3)) (Z 3 , +). Data from
Question:
Referring to Example 50.9, show that G(Q(3√2, i√3)/Q(i√3)) ≈ (Z3, +).
Data from in Example 50.9
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- Let 2 be the real cube root of 2, as usual. Now x³ 2 does not split in Q(V2), for Q(³2) < R and only one zero of x³ - 2 is real. Thus x³ - 2 factors in (Q(2)) [x] into a linear factor x 2 and an irreducible quadratic factor. The splitting field E of x³ - 2 over Q is therefore of degree 2 over Q(2). Then [E : Q] = [E : Q(√2)][Q(√2) : Q] = (2)(3) = : 6. We have shown that the splitting field over Q of x³ - 2 is of degree 6 over Q. We can verify by cubing that √√2 √2-1 are the other zeros of x³ 2 in C. Thus the splitting field E of x³ 2 over Qis Q(√2, i√3). (This is not the same field as Q(√2, i, √√3), which is of degree 12 over Q.) ·1+i√3 2 and -1-i√3 2
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