Zero divisors of a polynomial

Dependencies:

  1. Degree of product of polynomials

A ring $R$ has no zero-divisors iff $R[x]$ has no zero divisors.

Proof

Since $R \subset R[x]$, $R[x]$ has no zero-divisors implies $R$ has no zero-divisors.

Suppose $R$ has no zero-divisors.

Let $a(x), b(x) \in R[x]$.

\begin{align} & a \neq 0 \wedge b \neq 0 \\ &\Rightarrow \deg(a) \ge 0 \wedge \deg(b) \ge 0 \\ &\Rightarrow \deg(a) + \deg(b) \ge 0 \\ &\Rightarrow \deg(ab) \ge 0 \tag{Since $R$ has no zero-divisors, $\deg(ab) = \deg(a) + \deg(b)$} \\ &\Rightarrow ab \neq 0 \end{align}

Therefore, $R[x]$ has no zero-divisors.

Dependency for:

  1. Product of linear factors is a factor
  2. Polynomial division theorem
  3. Gauss' Lemma

Info:

Transitive dependencies:

  1. Group
  2. Ring
  3. Polynomial
  4. Degree of product of polynomials