>>195 補足

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"Z(1)" nth roots of unity algebraic closure of Z/pZ
で、下記ヒット

where Z/nZ(1) is the group of n-th roots of unity in K￣
Z＾(1) := lim ←-n Z/nZ(1)

とある。＾（ハット）は、完備化でよく使われる
Z(1)を完備化していると思われる
Z(1)に、n-th roots of unityに含まれていれば
完備化したZ＾(1)にも入っているだろう

(参考)
https://arxiv.org/pdf/1603.05811.pdf
Finite and ´etale polylogarithms
Kenji Sakugawaa, Shin-ichiro Sekib,
aDepartment of Mathematics, Graduate School of Science Osaka University Toyonaka, Osaka 560-0043 Japan
bDepartment of Mathematics, Graduate School of Science Osaka University Toyonaka, Osaka 560-0043 Japan
Preprint submitted to Elsevier October 4, 2016

P3
1.5. Notation
Let K be a field of characteristic 0. We denote by μ(K) the set of roots of unity in
K. We fix an algebraic closure K￣ of K and the symbol GK denotes the absolute Galois
group Gal(K/K) of K. Let L be a local field or an algebraic extension of Q. Then,

we denote by OL the ring of integers of L. For each locally noetherian affine scheme
Spec(R) and for each topological abelian group A equipped with a continuous action of
the ´etale fundamental group π := πet1(Spec(R)) of Spec(R),
We denote by κn,K : K×/(K×) n 〜-→ H1(K, Z/nZ(1))
the Kummer map induced by the Kummer sequence
1 → Z/nZ(1) → K× n-→ K× → 1
where Z/nZ(1) is the group of n-th roots of unity in K￣

P4
2. Review of ´etale polylogarithms

We regard this coherent system as a basis of Z＾(1) := lim ←-n Z/nZ(1).