Isomorphism between $K[x]$ and $K[x_1, x_2, \dots, x_n]$

Question

n $\in \mathbb{N}$, n > 1, How to prove that there is isomorphism between $K[x]$ and $K[x_1, x_2, \dots, x_n]$ as vector spaces, but isn't isomorphism between $K[x]$ and $K[x_1, x_2, \dots, x_n]$ as ring's?

I think I need show reflection between $k$ and ${k_1,k_2,\dots,k_n}$ to make reflection $x^k$ in $x_1^{k_1}x_2^{k_2}\dots x_n^{k_n}$ and then I show that homomorphism between $K[x]$ and $K[x_1, x_2, \dots, x_n]$ as rings doesn't save multiplication and with homomorphism between $K[x]$ and $K[x_1, x_2, \dots, x_n]$ as vector space is all good. Is it right way to solve it? If it's all right, how to make good reflection between $k$ and ${k_1,k_2,\dots,k_n}$. Good for me is when we can get all possible combinations ${k_1,k_2,\dots,k_n}$.

Answer 1

Hint: $K[x]$ is a principal ideal domain. Can you think of an ideal that is not principal in $K[x,y]$?

Answer 2

The result is essentially this

Being that basis elements (generating sets) determine vector spaces up to isomorphism.

To see that $k[x]$ differs from the rest as a ring, you should find some property that it does not share with the rest (think about the analogy between this ring and the integers).