Question

In Exercises 5-8, each element of the given congruence-class ring can be written in the form$\left[ax+b\right]$ (Why?). Determine the rules for addition and multiplication of congruence classes. (In other words, if the product$\left[ax+b\right]\left[cx+d\right]$ is the class $\left[rx+s\right]$, describe how to find and from and similarly for addition.)

7.$\mathbf{\square }\left[x\right]\mathbf{/}\left(x^2-3\right)$

Short answer

The rules for addition and multiplication $\underline{are}\sout{is}$:

$$\begin{gathered} \left[a_0+a_{1}x\right]+\left[b_0+b_{1}x\right]=\left[\left(a_0+b_0\right)+\left(a_1+b_1\right)x\right] \\ \left[a_0+a_{1}x\right]+\left[b_0+b_{1}x\right]=\left[\left(a_0{b}_0+3a_1{b}_1\right)+\left(a_0{b}_1+3a_1{b}_0\right)x\right] \end{gathered}$$

Step-by-step solution

Finding rules for addition

Consider the given function,$\square \left[x\right]/\left(x^2-3\right)$

As $x^2\equiv 3\left(mod{x}^2-3\right)$,$\left[a_0+a_{1}x\right]$ this is the representation of equivalence class.

The addition table can be calculated as:

$\left[a_0+a_{1}x\right]+\left[b_0+b_{1}x\right]+\left[\left(a_0+b_0\right)+\left(a_1+b_1\right)x\right]$

Finding rules for multiplication

The multiplication table writes $\sout{as}$,

$$\begin{gathered} \left[a_0+a_{1}x\right]·\left[b_0+b_{1}x\right]=\left[a_0{b}_0+\left(a_0{b}_1+a_1{b}_0\right)x+a_1{b}_1{x}^2\right] \\ =\left[a_0{b}_0+\left(a_0{b}_1+a_1{b}_0\right)x+a_1{b}_{1}3\right] \\ =\left[\left(a_0{b}_0+3a_1{b}_1\right)+\left(a_0{b}_1+a_1{b}_0\right)\right] \end{gathered}$$

Thus, the rules for multiplication is $\left[a_0+a_{1}x\right]·\left[b_0+b_{1}x\right]=\left[\left(a_0{b}_0+3a_1{b}_1\right)+\left(a_0{b}_1+a_1{b}_0\right)x\right]$.