Extensions of a field by a root of a monic irreducible polynomial

Question

Let $K$ be a field and $f(x) \in K[x]$ a monic irreducible polynomial of degree $n$ and consider $\frac{K[X]}{(f(x))}$.

Is $\frac{K[X]}{(f(x))} \cong K(a)$, $\forall a \in K$ st. $f(a)=0$ and $[K(a):K] = n$?

Answer 1

Let $a$ be a root of $f$. Since $f$ is irreducible, it is the minimal polynomial of $a$ (the polynomial of minimum degree with coefficients in $K$ having $a$ as a root). So no nonzero linear combination of $1, a, a^2, \dots a^{n-1}$ can equal $0$, so $K(a)$ has $K$-basis $1, a, a^2, \dots a^{n-1}$. Thus $[K(a) : K] = n$.

Similarly $K[X]/(f(X))$ is a vector space with basis $1, X, \dots X^{n-1}$ over $K$. The natural map $K[X]/(f(X)) \to K(a)$ sending $X^i \to a^i$ is therefore an isomorphism.