HOMEWORK 8

1. Determine the Galois group of the polynomial X^{4} + 1 over ℚ.

2. Determine the Galois group of the polynomial X^{4} + 4 over ℚ.

3. Let p be a prime integer, F a field, a ∈ F^{×}. Prove that the polynomial X^{p} − a is either split over F or irreducible.

4. Let F (α)/F be a field extension of prime degree p. Suppose that the minimal polynomial f of α over F has at least two roots in F (α). Prove that f is split over F (α).

5. Determine the Galois group of ℚ(\sqrt{6}, \sqrt{10}, \sqrt{14}, \sqrt{15}, \sqrt{21}, \sqrt{35})/ℚ.

6. Let K/F be a field extension of prime degree p and L/F a normal closure of K/F. Prove that [L : K] is not divisible by p.

7. Let f and g be nonconstant polynomials over a field F. Prove that Gal(fg) is isomorphic to a subgroup of Gal(f) × Gal(g).

8. Let f be a polynomial solvable by radicals. Show that the polynomial g(X) = f(X^{2}) is also solvable by radicals.

9. Prove that there is a cyclic field extension of ℚ of degree 1000.

10. Determine all prime integers p such that the congruence class [3] is a square in ℤ/pℤ.