Adjoining a root of a polynomial

Definition

Let $K$ be a field and let $f \in K[x]$ be irreducible. The field obtained by adjoining a root of $f$ to $K$ is the quotient ring $K[x]/(f)$. The map $K \to K[x]/(f)$, $c \mapsto c + (f)$, identifies $K$ with a subfield, and the class $\alpha = x + (f)$ is a root of $f$.

Remarks

The quotient is a field, because $f$ is irreducible. Its degree over $K$ is $\deg f$, with the basis $1, \alpha, \dots, \alpha^{\deg f - 1}$. For example, adjoining a root of $x^2 + 1$ to $\mathbb{R}$ gives a field isomorphic to $\mathbb{C}$.

Explanations

L1 human

You can create a root of a polynomial that has none. Compute with polynomials in $x$, and replace $f(x)$ by zero each time it appears. This is how $\mathbb{C}$ arises from $\mathbb{R}$ and $x^2 + 1$.