# Does this field contain i?

1. Feb 11, 2008

### antiemptyv

1. The problem statement, all variables and given/known data

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

2. Relevant equations

3. 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?

Last edited: Feb 12, 2008
2. Feb 12, 2008

### Hurkyl

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

3. Feb 12, 2008

### 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!