Homework Help: Extension fields

1. Mar 1, 2008

ehrenfest

[SOLVED] extension fields

1. The problem statement, all variables and given/known data
Let F be a field and let E be an extension field of F. Let $\alpha$ be an element of E that is algebraic over E. Is it true that all of the zeros of $irr(\alpha,F)$ are contained in the extension field $F(\alpha)$?
EDIT: I mean algebraic over F

2. Relevant equations

3. The attempt at a solution

Last edited: Mar 1, 2008
2. Mar 1, 2008

StatusX

Do you mean algebraic over F? And what is irr($\alpha$,F)?

3. Mar 1, 2008

ehrenfest

Yes. See the EDIT. irr(\alpha,F) is just the monic irreducible polynomial in F[x] that alpha is a zero of.

4. Mar 1, 2008

5. Mar 2, 2008

ehrenfest

It is false. Take (x^2+2)(x^2-2) over the rationals. Then zeros of the first factor are imaginary and the zeros of the second factor are real.

6. Mar 2, 2008

morphism

But (x^2+2)(x^2-2) is not irreducible over Q.

7. Mar 2, 2008

ehrenfest

Take x^3+3. It is irreducible by Eisenstein with p = 3. Its zeros are: $\sqrt[3]{3}e^{ik\pi/3}$ where k=0,1,2. Two of those values of k produce imaginary roots.

Last edited: Mar 2, 2008
8. Mar 2, 2008

morphism

That's better!