TITLE: How many roots does $f=x^4+x^2+2$ have in $\mathbb Q[x]/(f)$?
QUESTION [6 upvotes]: Given an irreducible $f\in \mathbb Q[x]$, how to determine the number of roots of $f=0$ in $\mathbb Q[x]/(f)$?
For some simple examples, I know how to do. If $f(x)=x^2+x-1$, the we use $x\in\mathbb Q[x]/(f)$ to represent $(-1+\sqrt 5)/2$, and $-x-1$ will be another root.
If $f=x^3-2$, $x\in\mathbb Q[x]/(f)$ will represent $2^{1/3}$, and it is the only root of $f=0$ in $\mathbb Q[x]/(f)$, since the other roots are complex.
But for more complicated $f$, e.g. $f=x^4+x^2+2$, I have no idea how to find the number of roots in the extended field. Thank you.

REPLY [3 votes]: In the case of $ \mathbf Q(2^{1/3}) $, one examines the situation locally, by embedding this field into a completion of $ \mathbf Q $ (in this case, $ \mathbf R $) and using what we know about $ \mathbf R $ to deduce that there can only be one root of $ X^3 - 2 $ in this number field. We may do something similar for the field $ \mathbf Q(\alpha) $, where $ \alpha $ is a root of $ X^4 + X^2 + 2 $. Indeed, we have the factorization 
$$ X^4 + X^2 + 2 = (X+2)(X+9)(X^2 + 5) \pmod{11} $$
which implies, by Hensel's lemma, that $ X^4 + X^2 + 2 $ has exactly $ 2 $ roots in the field $ \mathbf Q_{11} $ of $ 11 $-adic numbers. Now, we have an embedding $ \mathbf Q(\alpha) \to \mathbf Q_{11} $ by sending $ \alpha $ to either of these roots and extending; therefore, since $ \mathbf Q(\alpha) $ is then realized as a subfield of $ \mathbf Q_{11} $, it can also contain at most two roots of $ X^4 + X^2 + 2 $. We know that it contains at least two, since $ \alpha $ and $ -\alpha $ are distinct roots; therefore it contains exactly two such roots.