$[\mathbb{Q}(\theta):\mathbb{Q}] = \deg f$ for irreducible $f$

Let $f \in \mathbb{Q}[x]$ be irreducible of degree $d$, and let $\theta \in \mathbb{C}$ satisfy $f(\theta) = 0$. Then the subfield $\mathbb{Q}(\theta) \subseteq \mathbb{C}$ generated by $\theta$ has dimension $d$ as a vector space over $\mathbb{Q}$:

So every element of $\mathbb{Q}(\theta)$ is written uniquely as $\sum_{i<d} a_i\theta^i$ with $a_i \in \mathbb{Q}$, which is how GNFS represents number-field elements.