Let F be of characteristic p 0 We proved in class that an i

Let F be of characteristic p > 0. We proved in class that an irreducible f elementof F[x] is inseparable if and only if f (x) = g(x^p) for some g elementof F[x] (a) Prove that, in fact, any irreducible inseparable f (x) equals g(x^q) for some separable g and some q = p^n. (b) Use this to show that every root (in a splitting field) of an irreducible inseparable polynomial has the same multiplicity.

Solution

Given f , irreducible and inseparable.

By the remark preceding (done in the class), there exists a polynomial h in F[x] such that

                                                             f(x) = h(x^p)

Suppose h is not separable.(if it is separable , take g=h and we are done with q =p =p¹)

By applying the remark again (to h)

there exists a polynomial , say j(x) in F[x] with

                                              h(x) = j(x^p).

If j is separable, then             f(x) = j(x^q) with q =p²

Otherwise continue till we reach

                                              f(x) = g(x^q), with g separable and q a power of p.

(b) Let f(x) = g(x^q)

If a is a root then, g(a^q) = g(a)^q =0 implies that the multiplicity is the same.