# Extension fields

[SOLVED] extension fields

## Homework Statement

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

## The Attempt at a Solution

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

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

morphism
Homework Helper

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.

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

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

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morphism