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 $(3^{ed})$.
Best Answer
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 $f(D)=X=Y=0$.