Let $\pi:$(the 5-adic integers)$\to \mathbb{Z}/5\mathbb{Z}$ be the reduction map.

Let $f:\mathbb{Z}/5 \mathbb{Z} \to $ (the 5-adic integers) have the following properties $\forall x,y\in \mathbb{Z}/5\mathbb{Z}$

  1. $f(x)f(y)=f(xy)$

  2. $f(x)^{p-1}=1$ unless $f(x)=0$

  3. $\pi(f(x))=x$

Also, let us adjoin a primitive $5$th root of unity, call it $\zeta_5=\zeta$, to the 5-adic integers.

I want to find the 5-adic order of $\zeta-f(\overline 2)\zeta^2+f(\overline 2)\zeta^3+\zeta^4$

I see that the order of $1-\zeta^i$ is $\frac 1 4$ for $i\not\equiv0$ (mod 5).

I also see that $f(\overline 4)=-1$ and that therefore $f(\overline 2)^2=-1$ and $f(\overline 3)^2=-1$. Perhaps this can be used to rewrite the sum $\zeta-f(\overline 2)\zeta^2+f(\overline 2)\zeta^3+\zeta^4$ in a "better" way.

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    $\begingroup$ I would certainly write $f(\overline 2)$ as good-old $i$, square root of $-1$. Then your expression is $\zeta+i^3\zeta^2+i\zeta^3 +\zeta^4$. $\endgroup$ – Lubin Dec 18 '18 at 2:19
  • $\begingroup$ @Lubin Certainly makes notation a lot easier to work with. Thanks for the idea. I feel like multiplying by some $i^aζ^b$ term would help. $\endgroup$ – Pascal's Wager Dec 18 '18 at 2:25
  • $\begingroup$ Using a symbolic algebra package, I seem to find that your number is a unit in its ring. Does that seem right, or wrong to you? $\endgroup$ – Lubin Dec 18 '18 at 2:33
  • $\begingroup$ @Lubin I think that's right. $\endgroup$ – Pascal's Wager Dec 18 '18 at 2:36
  • $\begingroup$ Then I suggest defining $\sigma$ in the Galois group over $\Bbb Q_5$, $\sigma(\zeta)=\zeta^2$, it’s a generator, and taking the Norm of your element, which just means replacing $\zeta$ by $\zeta^2$, $\zeta^3$, and $\zeta^4$ in your expression, and multiplying all four four-term factors together. You’ll probably get a lot of collapsing, to make the result obvious. (I’m displaying my ignorance of group-representation theory here, of course. To those who are not so ignorant, your result is probably obvious.) $\endgroup$ – Lubin Dec 18 '18 at 2:41

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