The rings $\mathbb{Z}/p\mathbb{Z}$, $\mathbb{Z}\left[\frac{1}{p}\right]$, ${\mathbb{Z}}_{\left(p\right)}$ and ${\mathbb{Z}}_p$

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 16 May 2012

The rings $\mathbb{Z}/p\mathbb{Z}$, $\mathbb{Z}\left[\frac{1}{p}\right]$, and ${\mathbb{Z}}_{\left(p\right)}$

Let $p\in {\mathbb{Z}}_{>0}$ be a prime (irreducible element of $\mathbb{Z}$).

• The $p$-clock is the ring $$\mathbb{Z}/p\mathbb{Z}=\left\{m\right|m\in \mathbb{Z}\}\text{with}m=n\text{if}m-n\in p\mathbb{Z},$$ where $p\mathbb{Z}=\left\{p\ell \right|\ell \in \mathbb{Z}\}$ is the set of multiples of $p$ in $\mathbb{Z}$. Temporarily write $\left[m\right]$ for the element $m$ in $\mathbb{Z}/p\mathbb{Z}$ so that we may distinguish $\mathbb{Z}/p\mathbb{Z}$ from $\mathbb{Z}$. With this notation, the addition and multiplication in the ring $\mathbb{Z}/p\mathbb{Z}$ are given by $$\left[m\right]+\left[n\right]=[m+n]\text{and}\left[m\right]\left[n\right]=\left[mn\right].$$
• The rings $\mathbb{Z}\left[\frac{1}{p}\right]$ and ${\mathbb{Z}}_{\left(p\right)}$ are the subrings of $\mathbb{Q}$ given by $$\mathbb{Z}\left[\frac{1}{p}\right]=\left\{\frac{m}{p^k}\right|k\in {\mathbb{Z}}_{\ge 0},m\in \mathbb{Z}\}\text{and}{\mathbb{Z}}_{\left(p\right)}=\left\{\frac{m}{n}\right|m,n\in \mathbb{Z},n\ne 0,n\text{is prime to}p\}.$$

The ring ${\mathbb{Z}}_p$

The real numbers $ℝ$ contain the integers $\mathbb{Z},$ $$\begin{array}{ccl}ℝ& =& {a_l{10}^l+a_{l-1}{10}^{l-1}+a_{l-2}{10}^{l-2}+\cdots |l\in \mathbb{Z},a_j\in \frac{\mathbb{Z}}{10\mathbb{Z}}} \\ \cup |& & \\ \mathbb{Z}& =& {a_l{10}^l+a_{l-1}{10}^{l-1}+\cdots +a_{1}10+a_0|l\in {\mathbb{Z}}_{\ge 0},a_j\in \frac{\mathbb{Z}}{10\mathbb{Z}}} \end{array}$$ where $\frac{\mathbb{Z}}{10\mathbb{Z}}= {0,1,2,3,4,5,6,7,8,9}$ and the addition and multiplication in $ℝ$ are compatible with the addition and multiplication in $\mathbb{Z}$.

The p-adic numbers ${\mathbb{Q}}_p$ contain the p-adic integers ${\mathbb{Z}}_p$ and the integers $\mathbb{Z},$ $$\begin{array}{ccl}{\mathbb{Q}}_p& =& {a_{-l}{p}^{-l}+a_{-l+1}{p}^{-l+1}+a_{-l+2}{p}^{-l+2}+\cdots |l\in \mathbb{Z},a_j\in \frac{\mathbb{Z}}{p\mathbb{Z}}} \\ \cup |& & \\ {\mathbb{Z}}_p& =& {a_0{p}^0+a_1{p}^1+a_2{p}^2+\cdots |a_j\in \frac{\mathbb{Z}}{p\mathbb{Z}}} \\ \cup |& & \\ \mathbb{Z}& =& {a_0{p}^0+a_1{p}^1+a_2{p}^2+\cdots |a_j\in \frac{\mathbb{Z}}{p\mathbb{Z}} \; and\; all\; but\; a\; finite\; number\; of\; the\; a_j \; are\; 0} \end{array}$$ where $\frac{\mathbb{Z}}{p\mathbb{Z}}= {0,1,2,...,p-1}$ and the addition and multiplication in ${\mathbb{Q}}_p$ and ${\mathbb{Z}}_p$ are compatible with the addition and multiplication in $\mathbb{Z}$.

The rational functions $ℂ\left(\right(t\left)\right)$ contain the formal power series $ℂ\left[\right[t\left]\right]$ and the polynomials $ℂ\left[t\right],$ $$\begin{array}{ccl}ℂ\left(\right(t\left)\right)& =& {a_{-l}{t}^{-l}+a_{-l+1}{t}^{-l+1}+a_{-l+2}{t}^{-l+2}+\cdots |l\in \mathbb{Z},a_j\in ℂ} \\ \cup |& & \\ ℂ\left[\right[t\left]\right]& =& {a_0+a_{1}t+a_2{t}^2+\cdots |a_j\in ℂ} \\ \cup |& & \\ ℂ\left[t\right]& =& {a_0+a_{1}t+a_2{t}^2+\cdots |a_j\in ℂ \; and\; all\; but\; a\; finite\; number\; of\; the\; a_j \; are\; 0} \end{array}$$ where $ℂ$ is the complex numbers and the addition and multiplication in $ℂ\left(\right(t\left)\right)$ and $ℂ\left[\right[t\left]\right]$ are compatible with the addition and multiplication in $ℂ\left[t\right].$

HW: Show that, in ${\mathbb{Q}}_7,$ $$\begin{array}{rcl}\frac{1}{2}& =& 4+3\cdot 7+3\cdot 7^2+3\cdot 7^3+3\cdot 7^4+\cdots \\ -1& =& 6+6\cdot 7+6\cdot 7^2+6\cdot 7^3+6\cdot 7^4+\cdots \\ -\frac{1}{6}& =& 1+1\cdot 7+1\cdot 7^2+1\cdot 7^3+1\cdot 7^4+\cdots \\ \frac{1}{8}& =& 1+6\cdot 7+1\cdot 7^2+6\cdot 7^3+1\cdot 7^4+6\cdot 7^5+\cdots \end{array}$$

HW: Show that, in $ℂ\left(\right(t\left)\right)$, $$\begin{array}{rcl}\frac{1}{1-t}& =& 1+t+t^2+t^3+t^4+\cdots \\ e^t& =& 1+t+\frac{1}{2!}{t}^2+\frac{1}{3!}{t}^3+\frac{1}{4!}{t}^4+\cdots \\ \sin t& =& t-\frac{1}{3!}{t}^3+\frac{1}{5!}{t}^5-\frac{1}{7!}{t}^7+\cdots \\ \frac{1}{t^3(1-t)}& =& t^{-3}+t^{-2}+t^{-1}+1+t+t^2+\cdots \end{array}$$

The p-adic integers ${\mathbb{Z}}_p$ and the p-adic numbers ${\mathbb{Q}}_p$

Let $p\in {\mathbb{Z}}_{>0}$ be prime.

The p-adic valuation on $\mathbb{Q}$ is $v_p:{\mathbb{Q}}^{\times }\to \mathbb{Z}$ given by $$v_p\left(x\right)=l$$ if $x=p^l{p}_1^{l_1}\cdots p_r^{l_r}$ in its prime factorisation.

The fractional ideals of $\mathbb{Q}$ are the sets $$\left(p^m\right)= {x\in \mathbb{Q}|x=0 \; or\; v_p\left(x\right)\ge m} ,for\; m\in \mathbb{Z}.$$

The p-adic topology on $\mathbb{Q}$ is the topology generated by ${\left(p^m\right)|m\in \mathbb{Z}}$ as a system of fundamental neighborhoods of 0.

The p-adic numbers are the elements of ${\mathbb{Q}}_p$, the completion of $\mathbb{Q}$, wich respect to the p-adic topology, where the completion of a commutative topological group $G$ with fundamental system of neighborhoods $𝒩$ of 0 is the completion with respect to the uniformity on $G$ generated by the sets $$U_N= {\left(x,y\right)\in G\times G|y-x\in N} ,for\; N\in 𝒩.$$

The p-adic integers are the elements of ${\mathbb{Z}}_p$, the closure of $\mathbb{Z}$ in ${\mathbb{Q}}_p$.

HW: Show that ${\mathbb{Z}}_p$ is compact and open in ${\mathbb{Q}}_p$.

HW: Show that ${\mathbb{Z}}_p$ is a PID.

HW: Show that $p{\mathbb{Z}}_p$ is the unique prime ideal of ${\mathbb{Z}}_p$ and $$\frac{{\mathbb{Z}}_p}{p{\mathbb{Z}}_p}\simeq {𝔽}_{p}and\frac{p^m{\mathbb{Z}}_p}{p^n{\mathbb{Z}}_p}\simeq \frac{\mathbb{Z}}{p^{n-m}\mathbb{Z}},for\; m,n\in {\mathbb{Z}}_{>0} \; with\; m<n.$$

HW: Let $\left(E_n,f_{nm}\right)$ be the projective system indexed by ${\mathbb{Z}}_{>0}$ given by $$E_n=\frac{\mathbb{Z}}{p^n\mathbb{Z}}and\begin{array}{rrcl}{f}_{nm}:& \frac{\mathbb{Z}}{p^m\mathbb{Z}}& \to & \frac{\mathbb{Z}}{p^n\mathbb{Z}}\\ & a+p^m\mathbb{Z}& \mapsto & a+p^n\mathbb{Z}\end{array}for\; n\le m.$$ Show that $${\mathbb{Z}}_p\simeq \underleftarrow{\lim }\frac{\mathbb{Z}}{p^n\mathbb{Z}}.$$

HW: Show that ${\mathbb{Q}}_p$ is the field of fractions of ${\mathbb{Z}}_p$.

HW: Show that ${\mathbb{Z}}_p$ is the completion of $\mathbb{Z}$ in the $p$-adic topology.

HW: $ℝ$ is another completion of $\mathbb{Q}$. Why can't $ℝ$ be written as the field of fractions of an inverse limit?

Notes and References

This section mostly follows [Bou, Top Gen III §6 Ex 23]. [Se, Ch.II], [AM, Ch.10], [Bou, Top Gen III §6 Ex 23-26] and [Bou, Top Gen III §7 Ex 1] are fundamental references. The book [Gou] provides a book length exposition.

References

[AM] M. Atiyah and I.G. Macdonald, Introduction to commutative algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont. 1969 ix+128 pp. MR0242802.

[Bou] N. Bourbaki, General Topology, Springer-Verlag, 1989. MR1726779.

[Gou] MR1488696 (98h:11155) F.Q. Gouvêa, p-adic numbers. An introduction Second edition, Universitext, Springer-Verlag, Berlin, 1997. vi+298 pp. ISBN: 3-540-62911-4 MR1488696.

[Se] J.-P. Serre, A course in arithmetic, Translated from the French, Graduate Texts in Mathematics, No. 7, Springer-Verlag, New York-Heidelberg, 1973. viii+115 pp, MR0344216.

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