Let f be a field of characteristic p Prove that px xp a ei

Let f be a field of characteristic p. Prove that p(x) = x^p - a either is irreducible over F or splits in F.

Solution

Let E F be a splitting field for f, and let E be a root of f.

Then we have ^p = a, so

f = x^p-a = x^p- ^p = (x ) ^p .

Let f = g₁ · · · g_k be a factorization of f in F[X] into monic irreducible factors.

Each g_i is irreducible, monic, and has as a root.

Thus, each g_i = m_F,, so deg(m_F,) divides p.

This shows that either f = m_F, or each m_F, = x .

In the first case, f is irreducible over F and in the second case, F, so f splits over F.

Hence proved.