Thread: Roots and Irreducible polys

Any ideas?
f(x) belongs to F[x] and alpha is root of f(x) is some field extension of F. Show irreducible poly. g(x) belonging to F[x] has property that h(x) belonging to F[x] also has alpha as a root then g(x)/h(x) in F[x].

Let be a field and be its extension field. Let be algebraic over (how do we know it is algebraic?). Next define the evaluation homomorphism:
. We know that the is a principal ideal. So for some . This polynomial must be of minimial degree because otherwise if is smaller degree then we require that for some because the ideal is principal. But that is a contradiction because . Thus, is of minimal degree. Next we claim that is irreducible because if not and so one of the polynomial has the property that (WLOG) but that cannot be because has a smaller degree.

So we see that all the polynomial having this zero, i.e. are generated by an irreducible polynomial . And so if then for some . This shows that divides .

In our abstract class we have not yet been taught about kernels and ring homomorphisms so this has to be proved without using these. Thanks

Originally Posted by PacManisAlive

In our abstract class we have not yet been taught about kernels and ring homomorphisms so this has to be proved without using these. Thanks

1)Okay define .
2)Convince yourself that .
3)Since is a PID it follows for some .
4)Use a similar argument as above.

Note, this is basically the same approach except kernels eliminate the need to prove #1 which is a little lengthy but straightforward checking of the field axioms.