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

Last update: 16 May 2012

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

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

• The $p$-clock is the ring $ℤ/pℤ ={m|m∈ℤ} with m=nif m-n∈pℤ,$ where $pℤ=\left\{p\ell \phantom{\rule{.5em}{0ex}}|\phantom{\rule{.5em}{0ex}}\ell \in ℤ\right\}$ is the set of multiples of $p$ in $ℤ$. Temporarily write $\left[m\right]$ for the element $m$ in $ℤ/pℤ$ so that we may distinguish $ℤ/pℤ$ from $ℤ$. With this notation, the addition and multiplication in the ring $ℤ/pℤ$ are given by $[m]+[n] =[m+n] and [m][n] =[mn] .$
• The rings $ℤ\left[\frac{1}{p}\right]$ and ${ℤ}_{\left(p\right)}$ are the subrings of $ℚ$ given by $ℤ[1p] ={mpk | k∈ℤ≥0, m∈ℤ} and ℤ(p) ={mn | m,n∈ℤ, n≠0, nis prime top}.$

## The ring ${ℤ}_{p}$

The real numbers $ℝ$ contain the integers $ℤ,$ $ℝ = al 10l+ al-1 10l-1+ al-2 10l-2+⋯ l∈ℤ, aj∈ ℤ 10ℤ ∪| ℤ = al 10l+ al-1 10l-1 +⋯+ a110 +a0 l∈ ℤ≥0, aj∈ ℤ 10ℤ$ where $\frac{ℤ}{10ℤ}= {0,1,2,3,4,5,6,7,8,9}$ and the addition and multiplication in $ℝ$ are compatible with the addition and multiplication in $ℤ$.

The p-adic numbers ${ℚ}_{p}$ contain the p-adic integers ${ℤ}_{p}$ and the integers $ℤ,$ where $\frac{ℤ}{pℤ}= {0,1,2,...,p-1}$ and the addition and multiplication in ${ℚ}_{p}$ and ${ℤ}_{p}$ are compatible with the addition and multiplication in $ℤ$.

The rational functions $ℂ\left(\left(t\right)\right)$ contain the formal power series $ℂ\left[\left[t\right]\right]$ and the polynomials $ℂ\left[t\right],$ where $ℂ$ is the complex numbers and the addition and multiplication in $ℂ\left(\left(t\right)\right)$ and $ℂ\left[\left[t\right]\right]$ are compatible with the addition and multiplication in $ℂ\left[t\right].$

HW: Show that, in ${ℚ}_{7},$ $1 2 = 4+ 3⋅7+ 3⋅72+ 3⋅73+ 3⋅74+ ⋯ -1 = 6+ 6⋅7+ 6⋅72+ 6⋅73+ 6⋅74+ ⋯ - 1 6 = 1+ 1⋅7+ 1⋅72+ 1⋅73+ 1⋅74+ ⋯ 1 8 = 1+ 6⋅7+ 1⋅72+ 6⋅73+ 1⋅74+ 6⋅75+ ⋯$

HW: Show that, in $ℂ\left(\left(t\right)\right)$, $1 1-t = 1+ t+ t2+ t3+ t4+ ⋯ et = 1+ t+ 1 2! t2+ 1 3! t3+ 1 4! t4+ ⋯ sint = t- 1 3! t3+ 1 5! t5- 1 7! t7+ ⋯ 1 t3(1-t) = t-3+ t-2+ t-1+ 1+ t+ t2+ ⋯$

## The p-adic integers ${ℤ}_{p}$ and the p-adic numbers ${ℚ}_{p}$

Let $p\in {ℤ}_{>0}$ be prime.

The p-adic valuation on $ℚ$ is ${v}_{p}:{ℚ}^{×}\to ℤ$ given by $vp(x) =l$ if $x={p}^{l}{p}_{1}^{{l}_{1}}\cdots {p}_{r}^{{l}_{r}}$ in its prime factorisation.

The fractional ideals of $ℚ$ are the sets

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

The p-adic numbers are the elements of ${ℚ}_{p}$, the completion of $ℚ$, 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

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

HW: Show that ${ℤ}_{p}$ is compact and open in ${ℚ}_{p}$.

HW: Show that ${ℤ}_{p}$ is a PID.

HW: Show that $p{ℤ}_{p}$ is the unique prime ideal of ${ℤ}_{p}$ and

HW: Let $\left({E}_{n},{f}_{nm}\right)$ be the projective system indexed by ${ℤ}_{>0}$ given by Show that $ℤp ≃ lim ← ℤ pnℤ .$

HW: Show that ${ℚ}_{p}$ is the field of fractions of ${ℤ}_{p}$.

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

HW: $ℝ$ is another completion of $ℚ$. 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.