Properties of Degree

Theorem
Let $(R,+,\circ)$ be a ring with unity.

Let $R[X]$ be the ring of polynomial forms over $R$ in the indeterminate $X$.

For $f\in R[X]$ let $\deg(f)$ be the degree of $f$.

Then the following hold for all $f,g\in R[X]$:


 * $(1): \quad \deg(f+g)\leq\max\{\deg(f),\deg(g)\}$


 * $(2): \quad \deg(fg)\leq \deg(f)+\deg(g)$


 * $(3): \quad$ If $R$ is an integral domain then $\deg(fg)= \deg(f)+\deg(g)$

Proof
First we associate to $f=a_0+a_1X+\cdots+a_nX^n\in R[X]$ a formal vector $x_f=(a_0,a_1,\ldots,a_n,0,\ldots)\in R^\infty$, and let $x_f^i\in R$ denote the element at the $i^\text{th}$ position.

Then $\displaystyle \deg(f)=\sup\{i\in\N:x_f^i\neq 0\}$.

The sum $+$ and product $\circ$ in the polynomial ring $R[X]$ induce operations $+',\ \circ'$ on the subset $S=\{x\in R^\infty:x=x_f\text{ for some }f\in R[X]\}$. These are given by


 * $x_{f+g}^i=x_f^i+x_g^i$


 * $\displaystyle x_{f\circ g}^i=\sum_{j+k=i}x_f^jx_g^k$

Let $f,g\in R[X]$


 * 1. Let $d=\max\{\deg(f),\deg(g)\}$ Then $x_f^i=0=x_g^i$ for all $i>d$, so


 * $\displaystyle \sup\{i\in\N:x_{f+g}^i\neq 0\}=\sup\{i\in\N:x_f^i+x_g^i\neq 0\}\leq d$