F_2(α) isomorphic to F_2[x]/<f(x)>

[rukawakaede]

Homework Statement

Show that \mathbb{F}_2(\alpha) is isomorphic to \mathbb{F}_2[x]/ &lt;f(x)&gt;.

where f(x)\in\mathcal{F}_2[x] and E\supseteq\mathcal{F}_2 is an extension of \mathbb{F}_2 such that f(x) has a root \alpha\in E. Also \mathbb{F}_2(\alpha) is the subfield E generated by \mathbb{F}_2 and \alpha.

Homework Equations

The Attempt at a Solution

Could anyone give me some direction on how to start this proof?

 

[micromass]

What about the first isomorphism theorem?

Show that \mathbb{F}_2[X]\rightarrow \mathbb{F}_2[\alpha] is a surjection and find it's kernel...

 

[rukawakaede]

What about the first isomorphism theorem?

Show that \mathbb{F}_2[X]\rightarrow \mathbb{F}_2[\alpha] is a surjection and find it's kernel...

is it ok to say \mathbb{F}_2[\alpha]=\mathbb{F}_2(\alpha)?

 

[micromass]

is it ok to say \mathbb{F}_2[\alpha]=\mathbb{F}_2(\alpha)?

I'm sorry, I missed that. But now that you mention it, it seems that it is not correct what you're trying to prove: Take f(X)=(X-1)². Then F[X]/(X-1)² is not a field and can thus not be isomorphic to a field.