Timeline for p-adic expansion of roots of unity
Current License: CC BY-SA 4.0
20 events
| when toggle format | what | by | license | comment | |
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| Aug 14, 2019 at 16:58 | comment | added | Lubin | There isn’t a formula for the digits of a root of unity, any more than there’s a formula for the real digits of $\sqrt2$. But you can start with any integer $n$ prime to $p$, and go $z_0=n\mapsto n^p=z_1$ and then $z_n\mapsto z_n^p=z_{n+1}$. You get more digit with each iteration. The limit is the root of unity $\equiv n\pmod p$. | |
| Aug 14, 2019 at 16:46 | comment | added | wkm | Thanks for clarification, though I think it's a little bit hard to find $\omega_n$ explicitly, it depends on the prime $p$. | |
| Aug 14, 2019 at 16:35 | comment | added | reuns | Let $g$ an integer of order $p-1$ modulo $p$ and $w_n$ be the unique integer in $[0,p^n-1]$ that is $\equiv g^{p^n} \bmod p^n$, then $c_n = \frac{w_n-w_{n-1}}{p^n} \in 0 \ldots p-1$ and the $p$-adic series $w_0+\sum_{n=1}^\infty c_n p^n$ is equal to $\lim_{n \to \infty} g^{p^n}$ thus it is your primitive $p-1$ root of unity. | |
| Aug 14, 2019 at 16:32 | comment | added | wkm | I think I understand now, $w=g^{p^n}\mod p^n$ and we need to factor out $g^{p^n}$ to find the p-adic expansion of $\omega$ | |
| Aug 14, 2019 at 16:25 | comment | added | reuns | Not at all. I mean a p-adic a series is a p-adic limit, you need to understand why both are the same (ie. understand the definition of $\Bbb{Q}_p$) | |
| Aug 14, 2019 at 16:21 | comment | added | wkm | Do you mean $\omega=1+g^p+g^{p^2}+g^{p^3}+....$ | |
| Aug 14, 2019 at 16:17 | comment | added | reuns | To the reduction of $g^{p^n} \bmod p^n$, as for any $p$-adic series. | |
| Aug 14, 2019 at 16:16 | comment | added | wkm | so what are the $\alpha_i$ equal to ?? | |
| Aug 14, 2019 at 16:13 | comment | added | reuns | If $\omega$ is a $p-1$-primitive root then $\omega = \lim g^{a p^n}$ for some $a$ coprime with $p-1$. The RHS is a limit in the same way that a series such as $\sum_{n \ge 0} p^n$ is a limit. | |
| Aug 14, 2019 at 16:12 | comment | added | wkm | well if $\omega^n=1$ then I would like to write it as $\omega=1+\alpha_1p+\alpha_2p^2+..... $ | |
| Aug 14, 2019 at 16:11 | comment | added | reuns | What is not explicit to you ? | |
| Aug 14, 2019 at 16:11 | comment | added | wkm | Can you be more explicit please | |
| Aug 14, 2019 at 16:10 | history | edited | reuns | CC BY-SA 4.0 |
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| Aug 14, 2019 at 16:08 | comment | added | reuns | I gave it : it is $\zeta = g^{p^n} \bmod p^n$. Do you understand that $p$-adic series and $p$-adic limit is the same ? | |
| Aug 14, 2019 at 16:06 | comment | added | wkm | of the p-adic expansion of a given n-th root of unity as in David Lampert comment six minutes ago | |
| Aug 14, 2019 at 16:04 | history | edited | reuns | CC BY-SA 4.0 |
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| Aug 14, 2019 at 16:00 | comment | added | reuns | That's the general expression. The "coefficients" of what ? ${}{}$ | |
| Aug 14, 2019 at 15:55 | comment | added | wkm | I'm looking for the general expression of coefficients! | |
| Aug 14, 2019 at 14:10 | history | edited | reuns | CC BY-SA 4.0 |
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| Aug 14, 2019 at 14:05 | history | answered | reuns | CC BY-SA 4.0 |