Theorem. [1206]
Theorem. [1206]
Let $F$ be a field and $p(x)$ be an irreducible polynomial in $F[x]$. Then
- $K = F[x] / (p(x))$ is a field extension of $F$,
- $\theta = \overline{x} \in K$ is a root of $p(x)$ in $K$,
- $[K : F] = \deg p(x)$. Let $n = \deg p(x)$. Then $$1, \theta, \theta^{2}, \dots, \theta^{n-1}$$ is a basis of $K$ as an $F$-vector space. (Hence, $K = \{ a_{0} + a_{1} \theta + \cdots + a_{n-1}\theta^{n-1} \mid a_{0}, \dots, a_{n-1} \in F\}$.)