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Let $\mathbb Z$ be the ring of integers. We have the subring of $\mathbb Z[x]$ generated by integers and $p_1$ and $p_2$ ($p_1$ and $p_2$ are polynomials), and denote it as $\mathbb Z[p_1,p_2]$. I've got for my homework to investigate if $\mathbb Z[p_1,p_2]$ is UFD.
In the first task, $p_1=x^2-x^5$ and $p_2=x^2-2x^5$.
In the second task, $p_1=x^2+x^6$ and $p_2=x^2+2x^6$.

I don't even know how to start so if someone could help me I would be very grateful.

user26857
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Martin
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  • When in doubt, try simplifying or rewriting things. Can you find a simpler set of generators for $\mathbb{Z}[p_1,p_2]$? –  Apr 14 '13 at 18:40
  • Well, we can easily express x^2 and x^5 from p1 and p2 in first case. – Martin Apr 14 '13 at 18:45
  • I think this is relevant for the general case: https://math.stackexchange.com/questions/1591430/denominator-in-rational-gcd-of-integer-polynomials – lhf Apr 29 '20 at 17:07

1 Answers1

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I’ll attack the first case only. You’ve already seen that $R=\mathbb Z[x^2+x^5,\, x^2+2x^5]=\mathbb Z[x^2,\,x^5]$. Call $\xi=x^2$, $\eta=x^5$. Then of course $\xi^5=\eta^2$. I’ll leave it to you to show that $R\cong\mathbb Z[\Xi,H]/(\Xi^5-H^2)$, a ring in which $\Xi$ and $H$ both are indecomposables.

Lubin
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