Problem 9

Let $F$ be a field and let $I$ be a non-zero ideal in $F[x]$.

1. If $g(x)\in I$ is a non-zero polynomial of minimum degree, prove that $I=\left\langle g(x)\right\rangle$.

2. If $I=\left\langle g(x)\right\rangle$ for a $g(x)\in I$, prove that $g(x)$ is non-zero and of minimum degree.