Distinction between $\mathbb{K}(\alpha)$ and $\mathbb{K}[\alpha]$

Question

I apologize in advance wether the question is quite useless or silly but i'd like to clarify this concept once for all in my mind.

I'd like to know the difference between $\mathbb{K}(\alpha)$ and $\mathbb{K}[\alpha]$.

I know the statement that says "When $\alpha$ is algebric $\mathbb{K}(\alpha)$ = $\mathbb{K}[\alpha]$".

Besides that is quite obscure my knowledge on the subject.

Thanks

Answer 1

$K[\alpha]$ is the smallest ring containing $K$ and $\alpha$. Alternatively, you can think of it as all the polynomials in $\alpha$ with coefficients in $K$.

$K(\alpha)$ is the smallest field containing $K$ and $\alpha$. Alternatively, you can think of it as all the rational fractions in $\alpha$ with coefficients in $K$.