Let be of characteritic and let . Show that has no multiple roots and that is irreducible in if and only if for any .

, so which means is separable.

For the second part. If is irreducible then immediately it has no roots in . Conversely, suppose is a root of in some splitting field, is not in . Then by Freshman’s dream, one can verify are also roots of .

The irreducible polynomial of is a factor of . must contain all the roots of , otherwise the sum of these roots will results in . So is irreducible.

Jacoboson, Algebra I, p234.

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