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

$$\begin{array}{rl}& a\ne 0\wedge b\ne 0\\ & \Rightarrow \mathrm{deg}(a)\ge 0\wedge \mathrm{deg}(b)\ge 0\\ & \Rightarrow \mathrm{deg}(a)+\mathrm{deg}(b)\ge 0\\ \text{(Since }R\text{ has no zero-divisors, }\mathrm{deg}(ab)=\mathrm{deg}(a)+\mathrm{deg}(b)\text{)}& & \Rightarrow \mathrm{deg}(ab)\ge 0& \Rightarrow ab\ne 0\end{array}$$

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:

• Depth: 4
• Number of transitive dependencies: 4

Transitive dependencies:

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