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In the ring $\mathbb Z_5[X]$ find associated elements with $X^3+4X^2+3X+2$

I know that I must find $a \in \mathbb Z_5[X]$ such that $a|(X^3+4X^2+3X+2)$ and $(X^3+4X^2+3X+2)|a$. Assume that this $a$ exist we have:

$$\exists _{c\in \mathbb Z_5[X]} (X^3+4X^2+3X+2)=ac $$ $$\exists _{d\in \mathbb Z_5[X]} a=(X^3+4X^2+3X+2)d $$

However, I don't know how to find such $ a $ quickly

William8649
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  • Might be helpful $4X^2=-X^2$ and $2=-3$ – kingW3 Dec 31 '19 at 15:20
  • @kingW3 so I have: $X^3+4X^2+3X+2=X^3-X^2-2X+2=(X-1)(X^2-2)$ and divisors $X^3+4X^2+3X+2$ are $X-1$ and $X^2-2$. However this divisors have smaller degree than $X^3+4X^2+3X+2$ so we dont't have for example $X-1=(X^3+4X^2+3X+2)c$. So associated elements not exist? – William8649 Dec 31 '19 at 15:41
  • By the first dupe link, associates are unit multiples in a domain, and by the second link the only units in a polynomial ring over a field are the nonzero constants. – Bill Dubuque Dec 31 '19 at 19:29

1 Answers1

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Since $\mathbf Z/5\mathbf Z$ is a field, and for degree reasons in the polynomial ring over any integral domain, $a$ is necessarily one of $c(X^3+4X^2+3X+2)$, where $c\ne 0$.

Bernard
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  • Sorry, but I think your answer adds nothing to my answer, because you rewrote what I already wrote in my answer: "$\exists _{d\in \mathbb Z_5[X]} a=(X^3+4X^2+3X+2)d$" – William8649 Dec 31 '19 at 15:37
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    You didn't say in your question that $a$ is a multiple of $X^3+4X^2+3X+2$ by a (nozero) $\color{red}{\text{constant}}$. – Bernard Dec 31 '19 at 15:41