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!