HOMEWORK 3

1. Find an isomorphism between two finite fields 𝔽_{2}[X]/(X^{3} + X + 1)𝔽_{2}[X] and 𝔽_{2}[X]/(X^{3} + X^{2} + 1)𝔽_{2}[X].

2. Prove that ℚ(\sqrt{2 + \sqrt{2}})/ℚ is a normal extension.

3. Find a normal closure of the extension ℚ(2^{1/4})/ℚ and its degree over ℚ.

4. Find a normal closure of the extension ℚ(3^{1/3}, 3^{1/4})/ℚ and its degree over ℚ.

5. Let f ∈ ℚ[X] be an irreducible polynomial of degree 3. Assume that f has a unique real root α. Show that the extension ℚ(α)/ℚ is not normal.

6. Let F be a field of characteristic p. Show that the rational function field F (X) is a normal extension of F (X^{p}).

7. Let f be a nonzero polynomial over a field F and g be the greatest common divisor of f and its derivative f′. Prove that the polynomial f/g is separable.

8. Let F be a field of characteristic p > 0 and E/F an extension of degree relatively prime to p. Show that E/F is a separable extension.

9. a) Let E/F be a finite field extension. Show that the set of all elements in E, which are separable over F, is a subfield in E containing F. It is called the maximal separable subextension in E/F.

b) Let F be a field of characteristic 2. Find the maximal separable subextension in F (X)/F (X^{4} + X^{2}).

10. Let F be a field of characteristic p which is not a perfect field (i.e. F^{p} ≠ F). Show that F has a finite extension which is not separable.