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I'm doing Artin 11.5.1 and I got stuck on the definition of a residue. More specifically, the question 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 $(\alpha^3+\alpha^2+\alpha)(\alpha^5+1)$ in terms of the basis $[1,\alpha,\alpha^2,\alpha^3]$.

I know multiplication gives us the remainder of the product divided by $f$ - I have yet to try computing it - but what does "$\alpha$ is the residue of $x$" mean? Is it relevant to the solution?

user26857
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Lost
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1 Answers1

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There is a natural quotient map $\mathbb Z [x] \rightarrow \mathbb Z[x]/(f)$. The image of $x$ is called its residue.

Potato
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  • So, basically, I can just go ahead and do the multiplication? A straightforward computation? – Lost Dec 10 '13 at 07:31
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    @Lost There are some tricks, like $\alpha^5 + 1 = (\alpha^5 - 1) +2 \equiv 2$, so $(\alpha^3 + \alpha ^2 + \alpha)(\alpha^5 + 1) \equiv 2\alpha^3 + 2\alpha^2 + 2\alpha$ – tfw cant into math Dec 10 '13 at 07:36
  • @tfwcantintomath I actually just finished doing it myself, and found the same answer. Thanks for confirming my answer. As we skimmed over polynomial rings and adjoining elements, I don't have much intuition yet thus I want to make sure I'm doing it right. – Lost Dec 10 '13 at 07:38
  • @Lost Yes, it's just a straightforward computation. As tfw points out, you have some tricks available, too. – Potato Dec 10 '13 at 07:50