# Surjective Map between UFD and PID

Let $$k$$ be a field. Show that there does not exist a surjective ring homomorphism from $$k[x]$$ to $$k[x,y]$$.

I know that $$k[x]$$ is a Principal ideal Domain(PID) and $$k[x,y]$$ is a Unique Factorization Domain(UFD) but not PID.

Can Anyone help me to proceed further with the above knowledge, I have?
I am Reading Dummit and Foote $$(3^{ed})$$.

• Show $k[X,Y]$ is not a P.I.D., e.g. the ideal $(X, Y)$ is not principal – Bernard Jun 30 '19 at 14:16

Suppose that such a surjection $$f$$ exists, $$X=f(P), Y=f(Q)$$,
Let $$D=gcd(P,Q), P=AD, Q=BD$$ implies that $$X=f(A)f(D), Y=f(B)f(D)$$ implies that $$f(D)$$ divides $$X$$ and $$Y$$ and $$f(D)\in k$$.
$$D=UP+VQ$$ implies that $$f(D)=Xf(U)+Yf(V)$$ this is impossible since $$Xf(U)+Yf(V)$$ is constant only if it is $$0$$, this would implies $$f(D)=X=Y=0$$.