A question about polynomial rings This may be a trivial question.  We say an ideal $I$ in a ring $R$  is $k$-generated iff $I$ is generated by at most $k$ elements of $R$. Let $F$ be a field. Is it true that every ideal in $F[x_1,x_2,....,x_n]$ is $n-$generated. (This is true when $n=1$, because $F[x_1]$ is a PID)

Second question: Is it true that every ideal in $F[x_1,x_2,x_3,...]$ is generated by a countable set of elements of $F[x_1,x_2,x_3,...]$ ?

Thank you
The answer to the first question is no. For example, the ideal $(x^2, xy, y^2)$ in $F[x, y]$ cannot be generated by $2$ elements. To see this, note that any set of generators must linearly span the subspace of homogeneous quadratic polynomials, which has dimension $3$.
An ideal is finitely generated if it is finitely generated as $R$-modulo. So your first statement is wrong because there are at least the base-ring. For your second question note that Hilbert basis theorem say that if $R$ is a noetherian ring also $R[x]$ is a notherian ring. But you know that notherian rings are finitely generated as $R$-moduli.