HOMEWORK 2

1. Determine the degree of the extension ℚ(\sqrt{2},^{3}\sqrt{3}) over ℚ.

2. Let F = ℚ(\sqrt{2}, \sqrt{3}, \sqrt{5}) show that ^{3}\sqrt{2} ∉ F. (Hint: Show that the degree [F (^{3}\sqrt{2}) : ℚ] is divisible by 3.)

3. Determine the degree of the extension ℚ(\sqrt{3 + 2\sqrt{2}} ) over ℚ.

4. Determine the splitting field and its degree over ℚ for X^{4} − 2.

5. Determine the splitting field and its degree over ℚ for X^{6} − 4.

6. Prove that an algebraically closed field must be infinite.

7. Construct a field of 9 elements and give its addition and multiplication tables. Find a generator of the multiplicative group. How many generators are there?

8. Factor X^{8} − X into irreducibles in ℤ[X] and 𝔽_{2}[X].

9. Let f be an irreducible polynomial of degree n over a finite field 𝔽_{q}. Prove that f divides X^{q^{n}} − X.

10. Determine all q such that −1 is a square in 𝔽_{q}.