Math 6320, Assignment 3 Due in class: Tuesday, March 1 1. Determine Φ18 (x), i.e., the minimal polynomial of e2πi/18 over Q. 2. Determine the minimal polynomial of cos 2π/7 over Q. 3. If n is an odd positive integer, prove that Q(cos(2π/n)) = Q(cos(π/n)) . 4. Take a regular n-sided polygon inscribed in a circle of radius 1. Label the vertices P1 , . . . , Pn , and let λk be the length of the line joining Pk and Pn for 1 6 k 6 n − 1. Prove that λ1 · · · λn−1 = n . 5. Find a real number α such that the extension Q ⊂ Q(α) is Galois, with Galois group Z/6. 6. Determine the Galois group of xn − 1 over F p , where n is an integer that is not divisible by p. 7. Let m, n be positive integers. Determine Gal(F pmn /F pn ). 8. Express each of the following groups as a direct product of cyclic groups. Which of these arise (up to isomorphism) as multiplicative groups of fields? (a) (Z/6)× (b) (Z/8)× (c) (Z/18)× 9. Let α be the positive real number 51/4 . Prove that each of the extensions Q ⊂ Q(iα 2 ) and Q(iα 2 ) ⊂ Q(α + iα) is normal. Is Q ⊂ Q(α + iα) normal? 10. Let L = Q(e2πi/8 ). Determine all subgroups of Gal(L/Q), and the corresponding intermediate fields.