Let $K \subset \mathbb{C} $ be a field, and $K[x]$ the polynomial ring, $\alpha$ algebraic over $K$ and $L=K(\alpha)$.
I am reading this book and it states that, since any element of $L$ is of the form $\frac{f(\alpha)}{g(\alpha)}$ with $f,g \in K[x]$ and $g(\alpha) \neq 0$.
Considering $p=irr_k(\alpha)$ (the minimal polynomial of $\alpha$ over $K$), since $g(\alpha) \neq 0$ then $p \nmid g$. Here come my problem: Then they state that $<p,g> = K[x]$.
Why is this? I think this must come from a type of Bezout identity for polynomials, or something along that area. But I am unsure.
Your suspicion is correct. Bezout's identity holds in any Euclidean domain. Since $p$ is irreducible its only divisors are 1 and itself. Since $p$ does not divide $g$, their gcd must be 1.