Some conclusions on branch areas

references:

1. [书籍] Washington L C. Introduction to cyclotomic fields[M]. Springer Science & Business Media, 1997.
2. [Lecture notes] Milne J S. Algebraic number theory[M]. JS Milne, 2008.
3. [讲义] Milne J S. Fields and Galois Theory (v5. 10)[J]. Amer. Math. Monthly, 2021, 128(8): 753-754.
4. [SV11] Smart N P, Vercauteren F. Fully homomorphic SIMD operations[J]. Designs, codes and cryptography, 2014, 71: 57-81.
5. [GHS12a] Gentry C, Halevi S, Smart N P. Fully homomorphic encryption with polylog overhead[C]//Annual International Conference on the Theory and Applications of Cryptographic Techniques. Berlin, Heidelberg: Springer Berlin Heidelberg, 2012: 465-482.
6. [GHS12b] Gentry C, Halevi S, Smart N P. Homomorphic evaluation of the AES circuit[C]//Annual Cryptology Conference. Berlin, Heidelberg: Springer Berlin Heidelberg, 2012: 850-867.
7. [GHS12c] Gentry C, Halevi S, Smart N P. Better bootstrapping in fully homomorphic encryption[C]//International Workshop on Public Key Cryptography. Berlin, Heidelberg: Springer Berlin Heidelberg, 2012: 1-16.
8. [GHS12d] Gentry C, Halevi S, Peikert C, et al. Ring switching in BGV-style homomorphic encryption[C]//International Conference on Security and Cryptography for Networks. Berlin, Heidelberg: Springer Berlin Heidelberg, 2012: 19-37.
9. [GHPS12] Gentry C, Halevi S, Peikert C, et al. Field switching in BGV-style homomorphic encryption[J]. Journal of Computer Security, 2013, 21(5): 663-684.
10. [AP13] Alperin-Sheriff J, Peikert C. Practical bootstrap** in quasilinear time[C]//Annual Cryptology Conference. Berlin, Heidelberg: Springer Berlin Heidelberg, 2013: 1-20.
11. Algebraic number - Encyclopedia of Mathematics
12. Algebraic Number Theory Study Notes (1) - Algebraic Numbers and Algebraic Integers - Zhihu (zhihu.com)
13. [Abstract Algebra] 5. Permutation groups, simple groups, solvable groups, automorphism groups, free groups - Zhihu (zhihu.com)

Disclaimer: I have only learned the most elementary group ring field, and am not familiar with concepts such as fractional ideals, Galois expansions, and algebraic integer rings. This is just because when reading FHE papers, you will always encounter various concepts about divided rings, bits and pieces of knowledge picked out from books, papers, and web pages. (There should be a lot of mistakes.) Mathematics experts are welcome to give their advice, and other readers should also pay attention to identification.

Algebraic Integer

Algebraic structure:

• Rational number field $Q$, algebra closure ${\mathbb{Q}}^{ac}$, multiple range $\mathbb{C}$，满足关系 $Q\subseteq Q^{ac}\subseteq \mathbb{C}$
• Integer ring $WITH\subseteq Q$, algebraic integer ring$O_{\mathbb{C}}\subseteq Q^{ac}$

代数数（algebraic number）：元素 $a\in C$, there is a rational polynomial$f(x)\in \mathbb{Q}[x]$，使得 $f(\alpha )=0$, the set of all algebraic numbers is $Q^{ac}$，Construction area

Minimal polynomialminimal polynomial): algebraic number$a\in {\mathbb{Q}}^{ac}$, there is a unique leading irreducible polynomial$\varphi (x)\in \mathbb{Q}[x]$，满足 $\varphi (\alpha )=0$，使用 $n=\mathrm{ofg}\varphi$ nomological algebraic number $\alpha$ 的度数（degree）

an integer(algebraic integer):an integer $a\in Q^{ac}$, its minimal polynomial has integral coefficients$\varphi \left(x\right)\in Z\left[x\right]$, possessive algebra integer Set is $O_{\mathbb{C}}$, forming the whole ring

共轭（conjugates）：代数数 $a\in Q^{ac}$ 目极小多项表 $\varphi (x)$，它的所有根 $a_1,\cdots ,a_n$ are different from each other and are all algebraic numbers. Algebraic integers $a\in O_C$The conjugates of are all algebraic integers.

Divisibility (divisibility): I say algebraic integer $\beta$ Algebraic integer $a\ne 0$ Integer, existing algebraic integer $\gamma$, 得 $b=\gamma \alpha$，记文$\alpha |\beta$

单位（algebraic unit）：Our translation algebraic integer $e$ This is the first place, the result $1/e$ Also is an algebraic integer. One place is reversible (there is $O_C$superior)

associated (associated): two algebraic integers $\alpha ,\beta$, existential position $in\in O_C$，Use $a=ub$

扩域 $K/Q$, my story $\alpha$ Correct $Algebraic\; number\; on\; K$ if exists$f(x)\in K[x]$ 使得 $f(\alpha )=0$，similar definition $Concepts\; such\; as\; algebraic\; integers,\; minimal\; polynomials,\; conjugates\; on\; K$.

Integral closure (integral closure): Containing unitary commutative ring $R$ 的扩环$S$ 中，$The\; integer\; closure\; of\; R$ is $A\; subset\; of\; S$ where the elements $s\in S$ $R$ is an algebraic integer, that is, there is a leading polynomial$f(x)\in R\left[x\right]$ Use $f(s)=0$

Given a set of elements $a_1,\cdots ,a_n\in O_K$,and instead $⟨\alpha ,\beta ⟩=T{r}_{K/\mathbb{Q}}\left(\alpha \beta \right)$,square G = [ T r K / Q ( α 1 α 1 ) ⋯ T r K / Q ( α 1 α n ) ⋮ ⋱ ⋮ T r K / Q ( α n α 1 ) ⋯ T r K / Q ( α n α n ) ] G=\begin{bmatrix} Tr_{K/\mathbb Q}(\alpha_1 \alpha_1) & \cdots & Tr_{K/\mathbb Q}(\alpha_1\alpha_n)\\ \vdots & \ddots & \vdots\\ Tr_{K/\mathbb Q}(\alpha_n\alpha_1) & \cdots & Tr_{K/\mathbb Q}(\alpha_n\alpha_n)\\ \end{bmatrix} IndividualGram Difference
$$G=\left[\begin{array}{ccc}T{r}_{K/Q}({\alpha }_1{a}_1)& \cdots & T{r}_{K/Q}\left({\alpha }_1{a}_n\right)\\ ⋮& \ddots & ⋮\\ T{r}_{K/Q}({\alpha }_n{a}_1)& \cdots & T{r}_{K/Q}\left({\alpha }_n{a}_n\right)\end{array}\right]$$
The discriminant (discriminant) is defined as$d_{K/Q}({\alpha }_1,\cdots ,a_n)=\det (G)$，作用是 d K / Q ( α 1 , ⋯   , α n ) ≠ 0    ⟺    { α 1 , ⋯   , α n } d_{K/\mathbb Q}(\alpha_1,\cdots,\alpha_n) \neq 0 \iff \{\alpha_1,\cdots,\alpha_n\} dK/Q​(α1​,⋯,an​)=0⟺{ α1​,⋯,an​} 是 $Q$ is linearly independent.

If the number of expansions $[K:\mathbb{Q}]=n$，Nana integer $O_K$ Chichitei $The\; free\; Abelian\; group\; of\; n$: there is a set of basis$a_1,\cdots ,a_n\in O_K$,one $O_K=a_{1}WITH\oplus \cdots a_{n}Z$, we call it the integral base (Integeral base). The integral basis must exist, but it is not necessarily unique, but their discriminants are the same.

Cyclotomic Fields and Rings

令 $g_m$ is an abstract $m$ element (not necessarily $\mathbb{C},GF\left(p^d\right)$ (actual elements in ${the}_m=\exp (2\pi \sqrt{-1}/m)\in C$ is the primitive complex unit root,

• 分园多项式 ${Phi}_m(x)={\prod }_{i\in {\mathbb{Z}}_m^{\ast }}(x-{the}_m^i)\in \mathbb{Z}[x]$，是 $Irreducible\; polynomial\; on\; Q$, degree$n=\mathrm{ofg}{Phi}_m=\varphi (m)$

• $K=\mathbb{Q}({\zeta }_m)\cong \mathbb{Q}[x]/({\Phi }_m(x))$，The algebraic number of the elements

• 分园公司环$R=\mathbb{Z}[{\zeta }_m]\cong \mathbb{Z}[x]/({\Phi }_m(x))$, an algebraic integer is an algebraic integer

Domain expansion $K/Q$ This is one Galois 扩张 ∣ A u t ( K ) ∣ = [ K : Q ] = n |Aut(K)|=[K:\mathbb Q]=n < /span>, full $|Aut(K)|=[K:\mathbb{Q}]=n$ 以及 $Gal(K/\mathbb{Q})=Aut\left(K\right)$, possessive $n$ 个 $\mathbb{Q}$-自同构形如
$$t_i:g_m\in K\mapsto g_m^i\in K,\forall i\in {WITH}_m^{\ast }$$
迹（field trace）是 $\mathbb{Q}$-线性映射，
$$T{r}_{K/\mathbb{Q}}:a\in K\mapsto \sum _{i\in Z_m^{\ast }}{t}_i(a)\in Q$$
All linear mappings $L_r:K\to \mathbb{Q}$，都可以表示为 $L_r\left(a\right):=T{r}_{K/Q}(r\cdot a),\exists r\in K$

$K$ Work $C$ target child area, existence $n$ 个 $\mathbb{Q}$-内射环同态，称为嵌入（embedding），
$$p_i:g_m\in K\mapsto {the}_m^i\in \mathbb{C},\forall i\in {WITH}_m^{\ast }$$
Another expression of trace:
$$T{r}_{K/\mathbb{Q}}:a\in K\mapsto \sum _{i\in {\mathbb{Z}}_m^{\ast }}{p}_i(a)\in \mathbb{Q}$$
典范嵌入（canonical embedding）定义为
$$p:a\in K\mapsto ({\sigma }_i(a){)}_{i\in Z_m^{\ast }}\in {\mathbb{C}}^n$$

My defined area $The\; canonical\; embedding\; norm\; of\; K$ (Canonical Embedding Norm) is divided into $l_2$-范数、$l_{\infty }$-范数，令 $\Vert \cdot \Vert$ 是 $\mathbb{C}$ 的模长，
$$\begin{array}{rl}\Vert a{\Vert }_2^{can}& :=\Vert \sigma (a){\Vert }_2=\sqrt{\sum _i\Vert {\sigma }_i(a){\Vert }^2}\\ \Vert a{\Vert }_{\infty }^{can}& :=\Vert \sigma \left(a\right){\Vert }_{\infty }=\underset{i}{\max }\Vert {\sigma }_i(a)\Vert \end{array}$$
They satisfy good properties. Let $\Vert \cdot {\Vert }_l$ is a linear space $K/Q$ 向量的 $l$-范数，NA么

• $\Vert a\cdot b{\Vert }_2^{can}\le \Vert a{\Vert }_{\infty }^{can}\cdot \Vert b{\Vert }_2^{can},\forall a,b\in K$
• $\Vert a\cdot b{\Vert }_{\infty }^{can}\le \Vert a{\Vert }_{\infty }^{can}\cdot \Vert b{\Vert }_{\infty }^{can},\forall a,b\in K$
• $\Vert a{\Vert }_{\infty }^{can}\le \Vert a{\Vert }_1,\forall a\in K$
• $\Vert a{\Vert }_{\infty }\le c_m\cdot \Vert a{\Vert }_{\infty }^{can},\forall a\in K,\exists c_m=\Vert CR{T}^{-1}{\Vert }_{\infty }$

Tower of Cyclotomics

Thoughts $m^{\prime }|m$，简记 $K=\mathbb{Q}({\zeta }_m),R=\mathbb{Z}[{\zeta }_m]$ 以及 $K^{\prime }=Q\left({\zeta }_{m^{\prime }}\right),R^{\prime }=Z[{\zeta }_{m^{\prime }}]$，

1. Galois 扩张 $K/Q$，扩张order为 $n=\varphi (m)$，
   • 拥 $n$ self-same $t_i:K\to K$，Galois 群 $Gal(K/\mathbb{Q})\cong {WITH}_m^{\ast }$，
   • 拥 $n$ 个嵌入 $p_i:K\to \mathbb{C}$，典范嵌入 $p:K\to {\mathbb{C}}^n$
2. Galois 扩张 $K^{\prime }/Q$，扩张order为 $n^{\prime }=\varphi (m^{\prime })$，
   • 拥 $n^{\prime }$ $t_{i^{\prime }}^{\prime }:K^{\prime }\to K^{\prime }$，Galois 群 $Gal(K^{\prime }/\mathbb{Q})\cong {WITH}_{m^{\prime }}^{\ast }$，
   • 拥 $n^{\prime }$ 个嵌入 $p_{i^{\prime }}^{\prime }:K^{\prime }\to \mathbb{C}$，典范嵌入 $p^{\prime }:K^{\prime }\to C^{n^{\prime }}$

假设 $t=m/m^{\prime }$，then $n/n^{\prime }=\varphi (t{m}^{\prime })/\varphi (m^{\prime })=\varphi (t)$，从而：

• $K/K^{\prime }$ $\varphi \left(t\right)$ Next area, $K^{\prime }$ 视为 $K=K^{\prime }({\zeta }_m)$ Target area, area insertion $g_{m^{\prime }}\mapsto g_m^t$
• $R/R^{\prime }$ $\varphi \left(t\right)$ Next environment $R^{\prime }$ 视为 $R=R^{\prime }[{\zeta }_m]$ 目子环、环匌入为 $g_{m^{\prime }}\mapsto g_m^t$

Further, $K/K^{\prime }$ is also the Galois expansion, which forms a tower (tower),
$$\begin{array}{c}K\\ |\\ K^{\prime }\\ |\\ \mathbb{Q}\end{array}$$
它拥有 $\varphi (t)$ 个 $K^{\prime }$-automorphism, just those$i=1(\mathrm{mod}{m}^{\prime })$ 的 $\mathbb{Q}$-自同构，可验证
$$t_i({\zeta }_{m^{\prime }})=t_i\left({\zeta }_m^t\right)=g_m^{t(1+m^{\prime }\mathbb{Z})}=g_m^t=g_{m^{\prime }}$$
this is covered $\varphi (t)$-to-$1$ Transfer projection $i\in {WITH}_m^{\ast }\mapsto i\left(mod{m}^{\prime }\right)$ 诱导的、对应的self-construct关SYSTEMS $t_i\mapsto t_{i(mod{m}^{\prime })}^{\prime }$, they are in subdomain $K^{\prime }$ complete polymerization (coincide)

intermediate trace (intermediate trace) is $K^{\prime }$-线性映射，
$$T{r}_{K/K^{\prime }}:a\in K\mapsto \sum _{i=1\left(mod{m}^{\prime }\right)}{t}_i(a)\in K^{\prime }$$
易知 $T{r}_{K/\mathbb{Q}}=T{r}_{K^{\prime }/\mathbb{Q}}\circ T{r}_{K/K^{\prime }}$, and all linear mappings $L_r:K\to K^{\prime }$ 都可以表示为 $L_r\left(a\right):=T{r}_{K/K^{\prime }}(r\cdot a),\exists r\in K$

Similarly, by $\varphi \left(t\right)$-to-$1$ Transfer projection $i\in {WITH}_m^{\ast }\mapsto i\left(mod{m}^{\prime }\right)$ 诱导的、对应的Area Intrusion关SYSTEMS $p_i\mapsto p_{i\left(mod{m}^{\prime }\right)}^{\prime }$, they are also in subdomain $K^{Completely\; overlap\; on\; \prime }$. And it can be derived: for any $a\in K$ 和 $i^{\prime }\in {WITH}_{m^{\prime }}^{\ast }$，
$$p_{i^{\prime }}^{\prime }\circ T{r}_{K/K^{\prime }}:a\in K\mapsto \sum _{i=i^{\prime }(\mathrm{mod}{m}^{\prime })}{p}_i(a)\in K^{\prime }$$
我们把 $\sigma ,p^{\prime }$ is the integral of the infinitesimal,
$$p^{\prime }(T{r}_{K/K^{\prime }}(a))=P\cdot \sigma (a)$$
对应的矩阵 P ∈ { 0 , 1 } n ′ × n P \in \{0,1\}^{n' \times n} P∈{ 0,1}n′×n，系数为 $P_{i^{\prime },i}=1⟺i=i^{\prime }\left(mod{m}^{\prime }\right)$

Yu $The\; row\; vectors\; of\; P$ are orthogonal, and $l_2$-norm is exactly $\sqrt{n/n^{\prime }}$，可推出
$$\Vert T{r}_{K/K^{\prime }}(a){\Vert }_2^{can}=\Vert P\cdot \sigma (a){\Vert }_2\le \Vert a{\Vert }_2^{can}\cdot \sqrt{n/n^{\prime }}$$
i.e. linear mapping $T{r}_{K/K^{\prime }}$ generalshort element $a\in K$, mapped to anothershort element $T{r}_{K/K^{\prime }}\left(a\right)\in K^{\prime }$, the norm difference between the two is only a small factor.

Prime Split

order $A$ is one piece of Dedekind domain (Dedekind domain), $K$ is its fractional field (quotient field), finite divisible expansion$E/K$，令 $B$ Correct $A$ $E$ integral closure (integral closure), display $B=A[\alpha ],\exists \alpha \in E$，在令 $f(x)\in K\left[x\right]$ is small Multi-piece style.

• 令 $p\subseteq A$ 是素理想，令 $\bar{f}(x):=f\left(x\right)(\mathrm{mod}p)$，分解为
  $$\bar{f}\left(x\right)={\bar{P}}_1(x{)}^{{It\; is}_2}\cdot {\bar{P}}_1(x{)}^{{It\; is}_2}\cdots {\bar{P}}_g(x{)}^{{It\; is}_g}$$
  其中 ${\bar{P}}_i\left(x\right)\in (A/p)\left[x\right]$ This is a unique, unequivocal, multi-format

• My general ${\bar{P}}_i(x)$ 提升到 $P_i\left(x\right)\in A[x]$，定义理想
  $$p_i=(p,P_i(\alpha ))$$
  那么 $p_i$ Name $B$ 中的 lying over $p$ prime ideal, index ${It\; is}_i\ge 1$ is called the divergence index (ramification index), and has
  $$pB=p_1^{{It\; is}_1}{p}_2^{{It\; is}_2}\cdots p_g^{{It\; is}_g}$$
  This is the ideal $p$ present $B$ Intermediate decomposition. Results $\exists e_i\ge 2$，我们称 $p$ $E/K$中文歧（ramifie)

For cyclothematic domain $K=\mathbb{Q}({\zeta }_m)$，园环 $R=\mathbb{Z}[{\zeta }_m]$，素数 $p\in Z$ 对应的 $pR$ Possible $R$​ is decomposed into the product of prime ideal powers:

1. 计算 $m=\bar{m}\cdot p^k$, use $p\nmid \bar{m}$

2. 惯性度（inertial degree）：$d=ord(p\in {WITH}_{\bar{m}}^{\ast })$，

3. 分布天天（ramification index): $It\; is=\varphi (m)/\varphi (\bar{m})=\varphi (p^k)$，因为 Galois 扩张 ${It\; is}_1=\cdots ={It\; is}_g=It\; is$

4. Cyclic group ( p ) = { 1 , p , p 2 , ⋯ , p d − 1 } (p)=\{1,p,p^2,\cdots ,p^{d-1}\} (p)={ 1,p,p2,⋯,pd−1}，商群 $G={WITH}_{\bar{m}}^{\ast }/\left(p\right)$ target $f=\varphi (\bar{m})/d$，陪集 $i(p)$ 的代表 $i\in G$

5. Ideal $pR$ can be decomposed as follows,
   $$pR=\prod _{i\in G}{p}_i^e$$

6. The garden dividing polynomial is in $(\mathrm{mod}p)$ 下做分解 ${Phi}_{\bar{m}}(x)={\prod }_{i\in G}{F}_i\left(x\right)\left(modp\right)$，使用 $\mathrm{ofg}{F}_i=d$, then each prime ideal shape is as follows
   $$p_i=(p,F_i({\zeta }_m))$$
   They are different from each other, and their norms are all$|R/p_i|=p^d$

Because the prime ideals in the main ideal integral ring are maximum ideals, therefore$R/p_i\cong GF\left(p^d\right)$ are all the same finite field. Let ${In}_{\bar{m}}\in GF\left(p^d\right)$ is $\stackrel{Any\; element\; of\; ˉ}{m}$ (since $\bar{m}|p^d-1$, it must exist), we define ring homomorphism
$$h_i:g_m\in R\mapsto {In}_{\bar{m}}^i\in GF(p^d)$$
那么 $p_i$ します $\mathrm{because}{h}_i$, which induces domain isomorphism $h_i:R/p_i\to GF(p^d)$

For special cases:

• value $p$，下园环$\mathbb{Z}[{\zeta }_p]$ Intermediate ideal $(1-g_p)$ is prime, and $(1-g_p{)}^{p-1}=(p)$，Inko $p$ Current area $\mathbb{Q}({\zeta }_p)$ completely ramified (totally ramified)
• value $p$，它在 $Q({\zeta }_m$$⟺p|m$
• 如果 $p\nmid m$ (undivided), at this time $It\; is=1$，乘法阶 $d=ord(p\in {WITH}_m^{\ast })$，Zubunen area $Q\left({\zeta }_m\right)$ medium prime $p$ splitting $f=\varphi \left(m\right)/d$ The ideal cross.
• Further, if $p=1(\mathrm{mod}m)$，此时 $d=1$,square $p$ 完全分裂（splits completely），

$$pR=\prod _{i\in {\mathbb{Z}}_m^{\ast }}{p}_i$$

Power of Prime

令 $g_m$ is $m$ Secondary primitive unit root, divided garden number field$K=\mathbb{Q}[{\zeta }_m]$ Rationalization $R=O_K=\mathbb{Z}[{\zeta }_m]\cong \mathbb{Z}[x]/({\Phi }_m(x))$, the same reflection is $g_m\mapsto x$，它相对于 $Z$ 的扩张order $n=\varphi (m)$

对于Prime number $p\in Z$， ideal $pR$ can be decomposed into products of prime ideal powers: Calculation$m=\bar{m}\cdot p^k,p\nmid \bar{m}$，素数 $p\left(mod\bar{m}\right)$ 的乘法阶$d=ord(p\in {WITH}_{\bar{m}}^{\ast })$，商群 $G={WITH}_{\bar{m}}^{\ast }/\left(p\right)$ target $f=\varphi (\bar{m})/d$， index $It\; is=\varphi (m)/\varphi (\bar{m})=\varphi (p^k)$，
$$pR=\prod _{i\in G}{p}_i^e$$
Prime factorization of garden-partitioning polynomials${Phi}_{\bar{m}}(x)={\prod }_{i=1}^l{F}_i\left(x\right)(\mathrm{mod}p)$，
$$p_i=pR+F_i({\zeta }_m)R=(p,F_i({\zeta }_m))$$
Now generalize toprime powers $q=p^r\in \mathbb{Z}$，那么有
$$qR=\prod _{i=1}^l{p}_i^{re}$$
According to the Chinese Remainder Theorem,
$$R/qR\cong R/p_1^{re}\times \cdots \times R/p_l^{re}$$
Each small business ring $R/p_i^{re}$ (not a domain) is embedded in the entire ring ${WITH}_q$ (also not a domain) as a subring. The isomorphic mapping is:
$$a(modq)\mapsto (a(mod{p}_1^{re}),\cdots ,a(mod{p}_l^{re}))$$
预计mod-$q$ CRT set C = { c i } ⊆ R C=\{c_i\} \subseteq R C={ ci​}⊆R，满足
$$\begin{array}{rl}{c}_i& \equiv 1\left(mod{p}_j^{re}\right),j=i\\ c_i& \equiv 0\left(mod{p}_j^{re}\right),\forall j\ne i\end{array}$$
根据 CRT，逆映射就是
$$(a_1+p_1^{re},\cdots ,a_l+p_l^{re})\mapsto \left(\sum _{i=1}^l{a}_i\cdot c_i\right)+qR$$

Splitting in Cyclotomic Towers

对于 $m^{\prime }|m$，Branch Garden Entry Tower $R/R^{\prime }/Z$，素数 $p$的电影，

• 在 $R=\mathbb{Z}[{\zeta }_m]$ 上：计算 $m=\bar{m}\cdot p^k,p\nmid \bar{m}$, invariance $d=ord(p\in {WITH}_{\bar{m}}^{\ast })$，分见数数$It\; is=\varphi (m)/\varphi (\bar{m})=\varphi (p^k)$，商群 $G={WITH}_{\bar{m}}^{\ast }/\left(p\right)$ target $f=\varphi (\bar{m})/d$，
  $$pR=\prod _{i\in G}{p}_i^e$$

• 在 $R^{\prime }=Z[{\zeta }_{m^{\prime }}]$ 上：计算 $m^{\prime }={\bar{m}}^{\prime }\cdot p^{k^{\prime }},p\nmid {\bar{m}}^{\prime }$, inequality $d^{\prime }=ord(p\in {WITH}_{{\bar{m}}^{\prime }}^{\ast })$，分布数数${It\; is}^{\prime }=\varphi (m^{\prime })/\varphi ({\bar{m}}^{\prime })=\varphi (p^{k^{\prime }})$，商群 $G^{\prime }={WITH}_{{\bar{m}}^{\prime }}^{\ast }/\left(p\right)$ target $f^{\prime }=\varphi ({\bar{m}}^{\prime })/d^{\prime }$，
  $$p{R}^{\prime }=\prod _{i\in G^{\prime }}(p_{i^{\prime }}^{\prime }{)}^{{It\; is}^{\prime }}$$

令 $g:i\in G\mapsto i^{\prime }(\mathrm{mod}{\bar{m}}^{\prime })\in G^{\prime }$ This is one $f/f^{\prime }$-to-$1$ 的自然群同态（natural homomorphism）。素数 $p\in Z$ in $R^{\prime }$ is decomposed into a prime ideal$p_{i^{\prime }}^{\prime }\subseteq R^{The\; product\; of\; \prime }$, then each $p_{i^{\prime }}^{\prime }$ 在扩环 $R$ continues to decompose into prime ideal $p_i\subseteq R$ 的乘积，
$$p_{i^{\prime }}^{\prime }R=\prod _{i\in g^{-1}(i^{\prime })}{p}_i^{e/e^{\prime }}=\prod _{i=i^{\prime }(\mathrm{mod}\bar{m})}{p}_i^{e/e^{\prime }}$$
That is to say, every prime ideal$p_{i^{\prime }}^{\prime }\subseteq R^{\prime }$ are all decomposed into $f/f^{\prime }$ Discrete $R$ is the neutral prime ideal, which is based on $g:G\to G^{\prime }$ 对那些 lying oer $p$ 的素理想 $p_i\subseteq R$ partitioned.

给站环$R^{\prime }$ 目一组 mod-$p$ CRT set C ′ = { c i ′ ′ } ⊆ R ′ , ∣ C ′ ∣ = f ′ C'=\{c_{i'}'\} \subseteq R',|C'|=f' C′={ ci′′​}⊆R′,∣C′∣=f′, we choose the set S = { s j } ⊆ R , ∣ S ∣ = f / f ′ S=\{s_{ j}\} \subseteq R,|S|=f/f' S={ sj​}⊆R,∣S∣=f/f′，满足
$$\begin{array}{rl}{s}_j& \equiv 1(mod{p}_i^e),i=j\cdot {\bar{m}}^{\prime }+i^{\prime }\in G,\forall i^{\prime }\in G^{\prime }\\ s_j& \equiv 0\left(mod{p}_i^e\right),otherwise\end{array}$$
那么 Kronecker 积 $C=S\otimes C^{\prime }\subseteq R$ 下扩环 $R$ 目一组 mod-$p$ CRT set。考虑 cyclotomic tower，$R/R^{\prime }/\cdots /Z$, then the mod-$q$ CRT sets form a multiplicative structure. Easy to generalize to $q=p^r$ is the case of prime powers.