Let $R = \Bbb Z/6\Bbb Z = \{\bar 0,...\bar5\}$. In the polynomial ring $R[x]$ compute the product of the polynomials

Question

Let $R = \Bbb Z/6\Bbb Z = \{\bar 0,...\bar5\}$. In the polynomial ring $R[x]$ compute the product of the polynomials $f = \bar2x^2+\bar3x +\bar1\,,\, g = \bar3x^3 +\bar4x^2 +\bar2$

My attempt:

$$\begin{align}f\cdot g &=(\bar2x^2+\bar3x +\bar1)(\bar3x^3 +\bar4x^2 +\bar2)\\ &=(\bar6x^5+\bar9x^4+\bar8x^4)+(\bar3x^3+\bar{12}x^3+\bar4x^2)+(\bar4x^2+\bar6x+\bar2) \\&=\bar6x^5+\bar{17}x^4+\bar{15}x^3+\bar8x^2+\bar6x+\bar2\\ &=\bar0x^5+\bar5x^4+\bar3x^3+\bar2x^2+\bar0x+\bar2\\ &=\bar5x^4+\bar3x^3+\bar2x^2+\bar2 \end{align}$$

Is this correct? I'm pretty sure it is correct, but it's nice to double check.

Thanks!

Answer 1

Yes, your result is correct.

One way to check is to look at the product modulo $2$ and $3$.

I'll avoid the bars for some typing speed.

Modulo $2$, you have $f=x+1$ and $g=x^3$ so $fg=x^4+x^3$.

Modulo $3$, you have that $f=2x^2+1$ and $g=x^2+2$ so $fg=2x^4+2x^2+2$.

These agree with your result.