I'm doing an exercise in Stewart and Tall's Algebraic Number Theory which has me confused.
Suppose $\theta$ satisfies $p(\theta) = \theta^3 - 7\theta + 6 = 0$, but that $\theta \notin \Bbb R$. Find the degree of the field extension $\Bbb R(\theta)/\Bbb R$.
What is meant by $\theta \notin \Bbb R?$ The polynomial $p(x)$ splits entirely over $\Bbb R$ as $(x + 3)(x-2)(x-1)$ so that, considered over $\Bbb R$, the extension is trivial.. how should I think about $\theta$ when all of the roots of $p$ lie in $\Bbb R$?
This surely must be some typo: as you say, no such $\theta$ exists, since every root of $p(x)$ is real. I'm guessing the problem statement meant to have a slightly different cubic which has non-real roots, in which case the conclusion would be that $\mathbb{R}(\theta)$ is $\mathbb{C}$.