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Artin 11.5.1 asks

Let $f = x^4+x^3+x^2+x+1$ and let $\alpha$ be the residue of x in $R=\mathbb{Z}[x]/(f)$.
Express $g=(\alpha^3+\alpha^2+\alpha)(\alpha^5+1)$ in terms of the basis $(1,\alpha,\alpha^2,\alpha^3).$

I found a similar question here, but I'm not seeing connections to a couple things.

  • I know that there is a canonical map $\pi:\mathbb{Z}[x]\to\mathbb{Z}/(f)$ and that the residues are images of $\mathbb{Z}$ under $\pi$, but what do elements of $\mathbb{Z}/(f)$ look like?

  • While playing around with this problem, I divided $g$ by $f$ and obtained a remainder $2\alpha^3+2\alpha^2+2\alpha$. Was performing long division and identifying the remainder as the residue the intended goal of the problem?

  • In the link above there is a claim that $(x^5+1)=(x^5-1)+2\equiv 2$. How does that follow?

Community
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    The point is that $x^5 - 1$ is a multiple of $f$, and is therefore congruent to zero mod $f$. The answer to your first question is implicit in the statement that $(1,\alpha,\alpha^2,\alpha^3)$ forms a basis: every element of $R$ can be written uniquely in the form $a + b\alpha + c\alpha^2 + d\alpha^3$ where $a,b,c,d$ are integers. – user49640 Feb 20 '17 at 21:09

1 Answers1

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Since $\alpha^4+\alpha^3+\alpha^2+\alpha+1=0$, multiplying by $\alpha-1$ gives $\alpha^5-1=0$. Therefore $\alpha^5+1=2$. Thus $$ (\alpha^3+\alpha^2+\alpha)(\alpha^5+1)=2\alpha^3+2\alpha^2+2\alpha $$

egreg
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