Exercises VIII

1. A derivation of a ring R is a map D : R → R such that D(x + y) = D(x) + D(y) and D(xy) = xD(y) + D(x)y. Note that the zero map D(x) = 0 is a derivation of any ring.

(a) Show that if R = Z, R = Q, or R = Z/nZ then the only derivation of R is the zero derivation. (Hint: show that D(1) = 0 for any derivation of any ring.)

(b) Show that any derivation of Z[x] is of the form D = gd/dx for some g ∈ Z[x].

(c) Let f ∈ Q[x] be a polynomial and let R = Q[x]/(f). For which f does R have a nonzero derivation?

2. Let K = F (α)/F be a primitive algebraic extension of fields of characteristic p. A derivation of K over F is a derivation D of K such that D(a) = 0 for all a ∈ F. We use the notation Der(K/F) to denote the set of derivations of K over F.

(a) When does K have a nonzero derivation D over F?

(b) Consider the subfield F ⊂ L ⊂ K defined by the rule

L = {x ∈ K | D(x) = 0, ∀D ∈ Der(K/F)}.

Characterize L in terms of α and Irr(α, F).

3. Prove that

Q = Q((2 + \frac{10}{9}\sqrt{3})^{1/3} + (2 − \frac{10}{9}\sqrt{3})^{1/3}).

4. Show that in a finite field every element is the sum of two squares.

5. Let F be a field of characteristic p > 0, and let a ∈ F. Show that X^{p} − X + a either splits completely over F or is irreducible over F.

6. What is the Galois group of the polynomial 3x^{6} − 12x^{4} + 4x^{3} + 12x^{2} − 8 over Q? (This should be pretty tough! Feel free to use computer algebra to find the answer, but try to write up the arguments by hand. For example, write the roots as α_{1}, …, α_{6} and try to find expressions in them which are in Q to restrict the size of the Galois group.)