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Problem (Additional Properties Of Irreducible Polynomials Over A Field)

Let $F$ be a field and $p(x)$ an irreducible polynomial over $F$.

  1. Prove that $F[x]/\left<p(x)\right>$ is a field.
  2. Let $a(x),b(x)\in F[x]$. If $p(x)$ divides $a(x)b(x)$, then prove that $p(x)$ divides $a(x)$ or $p(x)$ divides $b(x)$.

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