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Let $f:\Bbb N^+\to\Bbb N^+/\langle2,3\rangle(\phi)$ replace the powers of $2$ and $3$ in the prime factorisation of any given positive integer with some number of the form $2^n\phi:n\in\Bbb N$, where $\phi$ is a sixth root of unity.

I've been using functions on the domain which use the 2-adic valuation and I want to define new functions on the range which use a new valuation that takes account of the roots of unity. Is there a natural such valuation* that extends the 2-adic value?

*Note I'm aware this isn't a field therefore this isn't technically a valuation but I used the word anyway as it's a subset of fields and I think it's clear what it means.

it's a hire car baby
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    If $\omega$ is a primitive cube root of unity, then all integers in the extended field have the form $a+b\omega$ where $a,b$ are ordinary $2$-adic integers. We can then render the valuation as $\max(|a|,|b|)$. – Oscar Lanzi Dec 09 '19 at 17:20
  • Thanks @OscarLanzi excuse me as I'm probably being really stupid, but am I right in thinking this doesn't value the primitive sixth roots of unity? There's probably some fact I'm overlooking through inexperience such as - those already have a valuation in $\Bbb Z_2$. – it's a hire car baby Dec 09 '19 at 17:39
  • Sorry @OscarLanzi I understand your comment better now. – it's a hire car baby Dec 09 '19 at 17:45
  • $-1 \in \Bbb{Z}_2$ and the sixth-root of unity are of the form $(-\omega)^n=a+b\omega$ with $(a,b)=(\pm 1,0),(0,\pm 1), \pm (1,1)$ of valuation $\min(\infty,0)=0$. – reuns Dec 09 '19 at 18:25
  • Thanks @reuns - the penny dropped shortly after my first reply! – it's a hire car baby Dec 09 '19 at 18:42
  • @reuns how is there a min in the evaluation? I thought it was the max of the component valuations. Also I understand all roots of unity should have, actually, unit valuation. – Oscar Lanzi Dec 09 '19 at 19:00
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    Since the minimal polynomial of a primitive sixth root of unity is $\Phi_6(x)=x^2-x+1$, this is actually a special case of what was attempted to be explained to you six months ago (https://math.stackexchange.com/a/3260194/96384), eighteen months ago (https://math.stackexchange.com/a/2840897/96384) and two years ago (https://math.stackexchange.com/a/2563837/96384). I must have missed something a year ago. See you next June. – Torsten Schoeneberg Dec 09 '19 at 19:00
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    Max of the absolute values, min of the valuations, $|a| = 2^{-v_2(a)}$. Together with Torsten comment it seems obvious you didn't understand the basic construction of $\Bbb{Z}_2= \varprojlim \Bbb{Z/2^n Z}$ and $\Bbb{Z}[\omega]=\Bbb{Z}[x]/(x^2+x+1)=\Bbb{Z}+\omega \Bbb{Z}$ – reuns Dec 09 '19 at 19:09
  • @reuns it looks like you were replying to Oscar's query thinking it was I. I understand the latter construction better but the first only vaguely, although my understanding of $\Bbb Z_2$ as a metric space is much better again. I knew this was a special case of my previous post. It was from those posts I learnt the method of appending a root of unity, although I have much more to learn. – it's a hire car baby Dec 10 '19 at 10:07

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