I'm going through a paper on homomorphic encryption by Smart and Vercauteren entitled "Fully Homomorphic SIMD operations" and had a question about some notation used in the paper. In section 2 of the above it is stated that for $F(X) \in \mathbb{F}_2[X]$, a monic polynomial of degree $N$ that splits into $r$ distinct irreducible factors of degree $d = N/r$ viz. $F(X)=\prod_{i = 1}^{r}F_i(X)$, $F(X)$ defines the field $\mathbb{K} = \mathbb{Q}(\theta) = \mathbb{Q}[X]/(F)$ where $\theta$ is a fixed root in the algebraic closure of the base field. Questions:
1) Should it not be stated that $\mathbb{Q}(\theta) \cong \mathbb{Q}[X]/(F)$ rather than $\mathbb{Q}(\theta) = \mathbb{Q}[X]/(F)$ ?
2) I assume that $\theta$ is just a root of $F(X)$ in the extension $\mathbb{Q}/ \mathbb{F}_2$?
3) In the expression $\mathbb{Q}[X]/(F)$, what is being referred to is the the quotient of $\mathbb{Q}[X]$ by the ideal generated by $F(X)$, i.e. $\mathbb{Q}[X]/(F(X))$?