Is i in the Field Generated by α Where α^3 + α + 1 = 0?

[antiemptyv]

Homework Statement

Is i \in \mathbb{Q}(\alpha), where \alpha^3 + \alpha + 1 = 0?

Homework Equations

The Attempt at a Solution

Suppose i \in \mathbb{Q}(\alpha). Then the field \mathbb{Q}(i) generated by the elements of \mathbb{Q} and i is an intermediate field, i.e.

\mathbb{Q} \subset \mathbb{Q}(i) \subset \mathbb{Q}(\alpha).

But the degree [\mathbb{Q}(i):\mathbb{Q}] = 2 does not divide the degree [\mathbb{Q}(\alpha):\mathbb{Q}] = 3, so i \notin \mathbb{Q}(\alpha).

Is that right?

 

[Hurkyl]

Sounds good. (You did verify that your polynomial is irreducible, right?)

 

[antiemptyv]

Great. Ahh, yes, that would certainly need to be shown. Thanks.

I just wanted to make sure I'm getting these basic ideas down correctly, and not missing something completely obvious. We're just beginning Galois theory, and I'm using a couple supplementary texts because the one we use in class (Algebra, Michael Artin) is a bit tough for a first exposure to this stuff. It's great, though, after you've got a good handle on things. Thanks again!