How to Use Euclidean Division in $\mathbb{Z}_7$?

[evinda]

Hello! :D
I am given the following exercise:
In $\mathbb{Z}_7$ apply the Euclidean division, dividing $[2]x^5+x^4+x^3+[3]x^2+[2]x+[2]$ by $[3]x^2+[2]x+[3]$.
That's what I have done:
$$\text{ the units of } \mathbb{Z}_7 \text{ are: } \{ 1,2,3,4,5,6\}$$
We want $[3]x^2+[2]x+[3]$ to be monic,so we multiply it by $[5]$ ($[3] \cdot [5]=1 \pmod 7$) and we get: $[5] \cdot ([3]x^2+[2]x+[3])=x^2+[3]x+[1] $.

Then ,dividing $[2]x^5+x^4+x^3+[3]x^2+[2]x+[2]$ by $x^2+[3]x+[1] $ I got, that the quotient is equal to $[2]x^3+[2]x^2+[1]$ and the remainder: $[6]x+[1]$.
Then, to make $[6]x+[1]$ monic,I multiplied it by $[6]$ : $[6]([6]x+[1])=x+[6]$ and dividing $x^2+[3]x+[1]$ by $x+[6]$,I found that the quotient is equal to $x+[4]$ and that the remainder is $[5]$.
Could you tell me if it is right or if I have done somethig wrong? (Blush)

 

[Opalg]

I agree with your calculation. So the conclusion is that those two polynomials are coprime over $\mathbb{Z}_7$.

 

[evinda]

I agree with your calculation. So the conclusion is that those two polynomials are coprime over $\mathbb{Z}_7$.

Great! (Clapping) Yes,since the greatest common divisor will be equal to $1$ (Nod) Thank you! :)