## Polynomial in a Field of Characteristic p

I’ll go ahead and it post it here as well:

Hey Ken, I’m trying to check if this proof is correct(it may not be the only way, but its one I thought of)

To Prove: Let be a field with . Let where . Show if is reducible, then has a root in (actually all).

This is actually an iff in the book, but this is the nontrivial direction.

My idea is the following. Let be a root of . Let be nonunit irreducible factors of . Now it is easy to see the set of all roots is the set . Thus, for some finite sets

Notice this means are both in . Let and . Then is a root for both and which are in . But since the degrees of these are ; respectively, the only way this is possible is if . But we could do this for any irreducible factor so each irreducible factor must have the same degree.

Hence, if with each irreducible, then for some . Since is prime either or must be . If then each root is in . If then is irreducible. Since is reducible the result follows.

After writing this out carefully it feels completely correct (of course I left out some easy details).