For example, finding the multiplicative inverse of$\displaystyle [x] in Z_5[x]/<x^2+x+1>$
or even something like
$\displaystyle [a+bx] in R[x]/<x^2+1>$
I cant seem to figure out this concept.
Thank you!
For example, finding the multiplicative inverse of$\displaystyle [x] in Z_5[x]/<x^2+x+1>$
or even something like
$\displaystyle [a+bx] in R[x]/<x^2+1>$
I cant seem to figure out this concept.
Thank you!
Divide $\displaystyle x^2+1$ by $\displaystyle bx+a$ with residue:
$\displaystyle x^2+1=(bx+a)(dx+c)+r$ , with $\displaystyle r=0\,\,\,or\,\,\,\deg(r)<1\Longrightarrow r\in\mathbb{R}\setminus{0}$.
As $\displaystyle x^2+1$ has no real roots, it can't be $\displaystyle r=0\Longrightarrow 0\ne r\in\mathbb{R}$ , and thus $\displaystyle (bx+a)\left(\frac{d}{r}x+\frac{c}{r}\right)=1\!\!\ !\pmod{x^2+1}\Longrightarrow (bx+a)^{-1}=\frac{d}{r}x+\frac{c}{r}\,\,\,in\,\,\,\mathbb{R }[x]\slash<x^2+1>$
Tonio