Question

Find a polynomial of degree 2 in${\mathbf{\mathbb{Z}}}_{\mathbf{6}}\left[x\right]$ that has four roots in role="math" localid="1648657713079" ${\mathbf{\mathbb{Z}}}_{\mathbf{6}}$. Does this

contradict Corollary 4.17?

Short answer

As${\mathrm{\mathbb{Z}}}_6$ is not a field, therefore, it is not a contradict corollary 4.17.

Step-by-step solution

Determine x2 + x is contradict or not

Consider the polynomial x²+ x,

Define $f\left(x\right)=x^2+x$to be the function associate of the assumed polynomial.

Now, it is proved that:

$$\begin{gathered} f\left(0\right)=0+0=0 \\ f\left(1\right)=1+1=2 \\ f\left(2\right)=4+2=0 \\ f\left(3\right)=3+3=0 \\ f\left(4\right)=4+4=2 \\ f\left(5\right)=1+5=0 \end{gathered}$$

Thus, x²+ x has four roots in ${\mathrm{\mathbb{Z}}}_6$.

Therefore, this is not a contradict corollary 4.17 because${\mathrm{\mathbb{Z}}}_6$ not a field.