If $K[X]\cong K[a]$ then $K(X)\cong K(a)$

Question

Let $L/K$ be a field extension. I came across the statement "if $a\in L$ is transcendent over $K$, then $K[X]\cong K[a]$, and thereby $K(X)\cong K(a)$".

Why is that?

Thanks in advance

Answer 1

If two integral domains are isomorphic, so are their quotient fields.

Answer 2

Hints:

$$\phi:K[X]\to K[a]\;,\;\;\phi p(X):=p(a)\;\;\text{is a ring isomorphism}$$

and thus we can extend it to::

$$\Phi: K(X)\to K(a)\;,\;\;\Phi\left(\frac{p(X)}{q(X)}\right):=\frac{p(a)}{q(a)}$$