Let be an imperfect field of characteristic . Prove that there exist irreducible inseparable polynomials over . Conclude that there exist inseparable finite extensions of .
Let be an element which is not a th power in , and let be a root of . In particular, . Now consider . Suppose is an irreducible factor of over . Now cannot be linear, since . But any factor of of degree at least 2 has a multiple root (since all the roots are ). So is an irreducible inseparable polynomial over .
Then is an inseparable extension of .