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

In summary, the conversation is about applying Euclidean division in $\mathbb{Z}_7$ and verifying if two polynomials are coprime over $\mathbb{Z}_7$. The process involves finding the units of $\mathbb{Z}_7$, making the divisor monic, dividing the dividend by the monic divisor, and checking the remainder and quotient. The conclusion is that the two polynomials are indeed coprime over $\mathbb{Z}_7$.
  • #1
evinda
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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)
 
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  • #2
I agree with your calculation. So the conclusion is that those two polynomials are coprime over $\mathbb{Z}_7$.
 
  • #3
Opalg said:
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! :)
 

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

What is the Euclidean division?

The Euclidean division is a mathematical process used to divide two numbers and find the quotient and remainder. It is based on the division algorithm developed by Euclid, a Greek mathematician.

How does the Euclidean division work?

The Euclidean division works by repeatedly subtracting the divisor from the dividend until the remainder is less than the divisor. The number of times the divisor is subtracted is the quotient, and the final remainder is the remainder of the division.

What is the purpose of using the Euclidean division?

The Euclidean division is used to find the exact quotient and remainder of a division, especially when dealing with large numbers. It is also used in various mathematical algorithms and proofs.

Can the Euclidean division be applied to all types of numbers?

Yes, the Euclidean division can be applied to all types of numbers, including integers, decimals, and fractions. It follows the same algorithm regardless of the type of numbers used.

Are there any limitations to using the Euclidean division?

There are some limitations to using the Euclidean division, such as when dividing by 0 or when dealing with irrational numbers. It is also not suitable for division involving complex numbers.

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