Prove that for every prime p, the polynomial x^p-x-1 is irreducible over Z_p
Don't even know where to start, can you please help me?
We will prove something stronger that if then is irreducible over . Notice that if (in some larger field) is a root of then is a root. This is because . Consequently, this means are the roots of (since degree is ). Therefore, is a splitting field over of . Now say that where are irreducible over . Let be any root of and be any root of . Remember by above and are splitting fields over of . Thus, . This means, obviously, that and so . In other words, all polynomials in the factorization of have the same degree. But a prime! Thus, either all are linear or else is its own irreducible polynomial factorization. It cannot be that all are linear for that would imply that the roots of lie in which is impossible because for all by Fermat's theorem. Thus, is irreducible.
Sure
A non-constant polynomial (where is a field, like ) is irreducible iff we cannot write where i.e. factorize into lower degree polynomials.
If is irreducible then has no zeros in . But the converse is not true. Consider . Certainly has no zeros in . However, . Thus, is reducible.
Does that make sense?