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. $$\deg(f+g)\leq\max\{\deg(f),\deg(g)\}$$


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


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

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


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


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

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


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