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So, I was doing an algebra exercise related to $\gcd$'s and $PID$'s and I need to check if some polynomials in $\mathbb{Z}_7[x]$ are units. The polynomials are the following:

\begin{equation*} g = x^2+5 \hspace{.5cm} \wedge \hspace{.5cm} h = x^2+6 \end{equation*} I know that if $g$ is a unit it must be invertible, i.e. \begin{equation*} \exists f \in \mathbb{Z}_7[x] \hspace{.15cm} | \hspace{.15cm}fg = 1 \end{equation*} but I can't seem to get one for any of the polynomials $f$ and $g$. Any help would be apreciatted.

xyz
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    The leading coefficient of $fg$ will be the product of the leading coefficients of $f$ and $g$, as this product is non-zero ($\mathbb{Z}_7$ has no zero divisors). Thus the product of two polynomials, not both constant, will be a non-constant polynomial. In particular it can never be $1$. – tkf Nov 22 '21 at 01:12
  • So basically when I am working in a $\mathbb{Z}_n[x]$ space, my units will only be polynomials of degree $0$? – xyz Nov 22 '21 at 01:14
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    No. In $\mathbb{Z}_4[x]$ we have $(1+2x)(1-2x)=1$. Can you see what the difference between the two cases is? – tkf Nov 22 '21 at 01:17
  • If $n$ is prime, my affirmation is valid? – xyz Nov 22 '21 at 01:17
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    Yes, exactly. $7$ is a prime, but $4$ is not. – tkf Nov 22 '21 at 01:18
  • Amazing explanation. If you want to elaborate a answer so I can check it, it would be my pleasure! :) – xyz Nov 22 '21 at 01:18
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    Thanks - I think a proof for the case of $\mathbb{Z}_n[x]$ would have been useful for people who have not studied more general rings, but the powers that be are keen to keep the site tidy and compact, so your question has been subsumed into a more general one. – tkf Nov 22 '21 at 01:32

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