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$$\deg(f+g)\leq \deg(f), \deg(g); \qquad \deg(f+g)<\deg(f),\deg(g) \iff \deg(f)=\deg(g). $$  However, if we multiply $f,g$, their product $fg$ can be outside $S$; for example  $$  f=x^2+1,\, g=x+1 \quad  \implies \quad  (x^2+1)(x+1)=x^3+x^2+x+1\,. $$  We describe a way to make $S$ "closed with respect to multiplication",that is, we want to define a special operation on the polynomials in $S$ that allows their product to \textbf{remain} in $S$.  We do that by what is called a "polynomial relation". For example fix example, we can define  the relations to be following relation  $x^3=x+1$, or equivalently $x^3+x+1=0$, as in the example above. We can limit the number of elements in $\Fb[x]$ in $x^3+x+1=0$.  With  this way: each relation in mind, any  time we find a monomial of degree greater than or equal to $3$ we substitute $x^3$ by with  $x+1$. At the end of We iterate  this process substitution until  we obtain anew  polynomial of degree strictly less than $3$. For example \[  x^4+x^2=x(x^3)+x^2=x(x+1)+x^2=x^2+x+x^2=x. x^5+x \,=\, x^2(\underline{x^3})+x \,=^{\mathrm{substitution}\, x^2(\underline{x+1})+x\,=\,  \underline{x^3}+x^2+x=^{\mathrm{substitution} \underline{x+1}+x^2+x \,=\, x^2+1 \,.  \]  One can obtain the same result dividing the polynomial $x^4+x^2$ by the polynomial $x^3+x+1$. The remainder is $x$.  Hence the polynomials defined in the set