Algebraic Field and polynomial

Question

Let K be a field and α be algebraic number. What is the difference between K(α) and K[α]? I couldn't find the definition and it seems to be used interchangeably.

Answer 1

$K[\alpha]$ is the integral domain generated by $K$ together with $\alpha$, which comes down to all polynomials in $\alpha$ with coefficients in $K$.

$K(\alpha)$ is the smallest field containing that $K[\alpha]$, so also includes the inverses of those polynomials, so we get elements of the form $\frac{p(\alpha)}{q(\alpha)}$, so the field of fractions essentially.

When $\alpha$ is algebraic over $K$, in fact $K(\alpha)=K[\alpha]$, (we can express all elements as degree $<n$ polynomials, where $n$ is the degree of the minimal polynomial of $\alpha$ over $K$).