Ring of polynomials become a field

Question

The K vector space K[α] is of finite dimension. Then The ring K[α] is field. Where α is an algebraic element.. I want an explanation and an example, i thought x doesn't have an inverse thus K[x] can never be a field just a ring, how come its a field here???

Answer 1

Let's take an example, maybe that will make things clearer.

Consider $\Bbb Q[\sqrt2]$. It is, in some sense, the polynomial ring over $\Bbb Q$ in the variable $\sqrt2$. A typical element looks like $$ 5\sqrt2^3-\frac75\sqrt2^2+2\sqrt2-\frac{140}{23} $$ However, it is also the case that $\sqrt2^2=2$, which means that the above can be simplified to $$ 12\sqrt2-\frac{1022}{115} $$ This simplification allows us to construct inverses to any non-zero element of $\Bbb Q[\sqrt2]$. For instance, the above element has inverse $$ \frac{13\,225}{2\,764\,316}\left(12\sqrt2+\frac{1022}{115}\right) $$