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- Jan 17, 2013

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This is Exercise 14 Chapter 3 Page 69

**Question**

*Let $G$ be the group of polynomials under the addition with coefficients from $Z_{10}$. Find the order of $f=7x^2+5x+4$ .*

Note: this is not the full question, I removed the remaining parts.

**Attempt**

For simplicity I represented the coefficients as ordered pair $(a,b,c)$

Then to compute $|f|$ , I add $f$ until I get a positive integer $n$ such that $f^n=(0,0,0)$

\(\displaystyle f^2=(7,5,4)+(7,5,4)=(4,0,8)\)

\(\displaystyle f^3=(4,0,8)+(7,5,4)=(1,5,2)\)

\(\displaystyle f^4=(1,5,2)+(7,5,4)=(8,0,6)\)

\(\displaystyle f^5=(8,0,6)+(7,5,4)=(5,5,0)\)

\(\displaystyle f^6=(5,5,0)+(7,5,4)=(2,0,4)\)

\(\displaystyle f^7=(2,0,4)+(7,5,4)=(9,5,8)\)

That approach seems wrong since that could go anywhere.

If we put $f=ax^2+bx+c$ then $f=(a,b,c)$ either circulates or approaches a finite order but what would be the method to prove it is of infinite or finite order ?