Show that for a field $F$, the polynomial ring $F[x_1, x_2, \ldots, x_n]$ is not a PID for $n>1$.

Question

I want clarification of the following solution:

Let $I=(x_1)+(x_2)$ be an ideal of $F[x_1, x_2, \ldots, x_n]$. Then if $I=(f)$ is principal then we must have $f \in F \backslash \{0\}$ since $\gcd(x_1, x_2)=1$ and $f \mid x_1,\ f \mid x_2$. Why?

But $I \cap F =\{0\}$. Why?

Contradiction, why?

Answer 1

Hints. $(x_1,x_2)=(f)\implies x_1\in(f),x_2\in(f)$.

If $a∈I∩F$ then $a=x_1g+x_2h$ and for $x_1=x_2=0$ we get...