HOMEWORK 4

1. Prove that X^{p−1} − 1 = (X − 1)(X − 2) · · · · · (X − p + 1) over 𝔽_{p}. Derive Wilson’s theorem: (p − 1)! ≡ −1 (mod p).

2. Prove that f(X)^{p} = f(X^{p}) for every polynomial f ∈ 𝔽_{p}[X].

3. Let E/F be a field extension of a field F of characteristic p > 0. Prove that for any α ∈ E, that is separable over F, F (α) = F (α^{p}).

4. Prove that a field F is perfect if and only if any finite extension of F is separable.

5. Prove that a finite extension of a perfect field is also perfect.

6. For a prime p and nonzero a ∈ 𝔽_{p} prove that X^{p} − X + a is irreducible and separable over 𝔽_{p}. (Hint: observe that if α is a root of X^{p} − X + a then α + 1 is also a root.)

7. Let F be a field of characteristic p. Prove that for every a ∈ F \ F^{p} the polynomial X^{p} − a is irreducible over F.

8. a) Let E/F be a finite field extension of a field F of characteristic p > 0. An element a ∈ E is called purely inseparable over F if a^{p^{n}} ∈ F for some n ≥ 0. Show that if a ∈ E is separable and purely inseparable over F, then a ∈ F.

b) A finite extension E/F is called purely inseparable if all elements in E are purely inseparable over F. Show that if E/F is a finite field extension, then E is purely inseparable over the maximal separable subextension in E/F.

9. Prove that a finite field extension E/F is purely inseparable if and only if any field homomorphism σ : F → L has at most one extension E → L.

10. Let F be a field of characteristic p > 0. Show that there are infinitely many fields K such that F (X^{p}, Y^{p}) ⊂ K ⊂ F (X, Y). (Hint: Consider fields F (X + f · Y) for different polynomials f ∈ F [X^{p}, Y^{p}].)