r
Q(G), where ζ7 is a primitive 7th root of 1 . Then E is the splitting...
8. Let n be a positive integer. The n-th cyclotomic polynomial Ф,a(z) E Z[2] is defined recursively in the following way: 1. Ф1(x)-x-1. 2. If n > 1, then Фп(x)- , (where in the product in the denomina- tor, d runs through all divisors of n less than n). . A. Calculate Ф2(x), Ф4(x) and Ф8(z): . B. n(x) is the minimal polynomial for the primitive n-th root of unity over Q. Let f(x) = "8-1 E Q[a] and ω...
Part D,E,F,G 10. Let p(x) +1. Let E be the splitting field for p(x) over Q. a. Find the resolvent cubic R(z). b. Prove that R(x) is irreducible over Q. c. Prove that (E:Q) 12 or 24. d. Prove: Gal(E/Q) A4 or S4 e. If p(x) (2+ az+ b)(a2 + cr + d), verify the calculations on page 100 which show that a2 is a root of the cubic polynomial r(x)3-4. 1. f. Prove: r(x) -4z 1 is irreducible in...