FALL 2005 #5

We show:

(1) [L: K] = 9.
(2) A simple extension of K by an element of L has degree at most 3.

(1) Note that x is a root of the polynomial z³ - x³ over F₃(x³, y)[z]. Now, any polynomial of the form z³ - a either splits or is irreducible over a field of characteristic 3. Since x is not in F₃(x³, y), it follows that z³ - x³ does not split; hence is irreducible over F₃(x³, y). Thus, L = F₃(x, y) = F₃(x³, y)(x) is a field extension of degree 3 over F₃(x³, y). Similarly, F₃(x³, y) = F₃(x³, y³)(y) is a degree 3 extension over K = F₃(x³, y³).

So [L: K] = [L: F₃(x³, y)][F₃(x³, y): K] = (3)(3) = 9.

(2) Suppose K(u) is a simple extension of K by an element u in L = F₃(x, y). Then

u = f(x, y) = a₁₁xy + . . . + a_ijxⁱy^j + . . . + a_nnxⁿyⁿ,

where each a_ij is an element of F₃.

Since we are in a field of characteristic 3, it follows that

u³ = f(x, y)³ = a₁₁³x³y³ + . . . + a_ij³x³ⁱy^3j + . . . + a_nn³x³ⁿy³ⁿ.

So u³ is an element of K = F₃(x³, y³). Thus, [K(u): K] is at most 3.