Question

List all associates of

(a)$x^2+x+1in{\mathrm{\mathbb{Z}}}_5\left[x\right]$

(b)$3x+2in{\mathrm{\mathbb{Z}}}_7\left[x\right]$

Short answer

Thus, the required answer is as follows:

(a) $x^3+x+1,2x^3+2x+2,3x^3+3x+3$and$4x^3+4x+4$ .

(b) $3x+2,2x+6,5x+1,x+3$and$4x+5$ .

Step-by-step solution

Monic associate 

It is given that a polynomial is$x^3+x+1$ and it has to find the associate in ${\mathrm{\mathbb{Z}}}_5\left[x\right]$.

Steps for Monic associate

(a). The set ${\mathrm{\mathbb{Z}}}_5\left[x\right]=\left\{1,2,3,4\right\}$. Then, the associate polynomial is written as:

$$\begin{gathered} p\left(x\right)=x^3+x+1 \\ p\left(x\right)=2x^3+2x+2 \\ p\left(x\right)=3x^3+3x+3 \\ p\left(x\right)=4x^3+4x+4 \end{gathered}$$

Hence, the associates are$x^3+x+1,2x^3+2x+2,3x^3+3x+3$ and $4x^3+4x+4$.

(b). The given polynomial is $3x+2$ . The set ${\mathrm{\mathbb{Z}}}_7\left[x\right]=\left\{1,2,3,4,5,6\right\}$. Then, the associate polynomial is written as:

$\begin{array}{rcl}p\left(x\right)& =& 3x+2\\ p\left(x\right)& =& 2\left(3x+2\right)\\ & =& 6x+4\end{array}$

For another associates,

$\begin{array}{rcl}p\left(x\right)& =& 3\left(3x+2\right)\\ & =& 9x+6\\ & =& 2x+6\end{array}$

$\begin{array}{rcl}p\left(x\right)& =& 4\left(3x+2\right)\\ & =& 12x+8\\ & =& 5x+1\end{array}$

$\begin{array}{rcl}p\left(x\right)& =& 5\left(3x+2\right)\\ & =& 15x+10\\ & =& x+3\end{array}$

$\begin{array}{rcl}p\left(x\right)& =& 6\left(3x+2\right)\\ & =& 18x+12\\ & =& 4x+5\end{array}$

Hence, the associates are$3x+2,2x+6,5x+1,x+3$ and$4x+5$ .