The p-Adic Numbers (Part I)

Image from Cover of Katok, 2007.

The motto of the Pythagoreans was “All is Number”. They saw numbers as the essence and foundation of the physical universe. For them, numbers meant the positive whole numbers, or natural numbers ${\mathbb{N}}$ , and ratios of these, the positive rational numbers ${\mathbb{Q}^{+}}$ . It came as a great shock that the diagonal of a unit square could not be expressed as a rational number.

If we arrange the rational numbers on a line, there are gaps everywhere. We can fill these gaps by introducing additional numbers, which are the limits of sequences of rational numbers. This process of completion gives us the real numbers $\mathbb{R}$ , which include rationals, irrationals like ${\sqrt{2}}$ and transcendental numbers like ${\pi}$ .

But there are other ways of completing ${\mathbb{Q}}$ that yield different number fields, one for every prime number ${p}$ . These are called the p-adic numbers, denoted ${\mathbb{Q}_p}$ . The p-adic numbers are essential in modern number theory: they are useful for solving diophantine equations and finding rational roots of polynomials. They have also been used in quantum mechanics and string theory. Other applications include probability theory, information theory, computer science and cryptography.

Norms and Metrics

The usual measure of the size of a rational number ${q\in\mathbb{Q}}$ is the absolute value ${|q|}$ . This also gives a measure of the distance between rational numbers, ${d(q_1,q_2) = |q_1-q_2|}$ , the Euclidean metric, which makes ${\mathbb{Q}}$ a metric space.

A Cauchy sequence ${\{a_n\}}$ is one in which `all terms beyond a certain point get close to each other’. Specifically, given any number ${\epsilon>0}$ there is some number ${N}$ such that the distance between any two terms beyond ${N}$ is less than ${\epsilon}$ . In symbolic form,

$\displaystyle \forall \epsilon>0,\ \exists N \in \mathbb{N} : \forall m,n>N,\ |a_m-a_n|<\epsilon$

We must be careful: the harmonic series

$\displaystyle a_1=1\,, \ \ a_2=1+\textstyle{\frac{1}{2}}\,, \ \ a_3=1+\textstyle{\frac{1}{2}}+\textstyle{\frac{1}{3}}\,, \dots$

is not a Cauchy sequence since, no matter how large ${N}$ may be, we can find two terms beyond ${a_N}$ that differ from each other by an indefinitely large quantity. All Cauchy sequences are bounded and converge in the real number system.

The definition of a Cauchy sequence depends crucially on the concept of distance. We use the Euclidean distance ${d(x,y)=|x-y|}$ defined by the absolute value. But it is possible to use other metrics and these lead to different completions of the rational numbers, known as p-adic numbers.

The p-adic Expansion of a Number

To construct the p-adic numbers, we define a new norm and a new measure of distance. This new norm appears at first to be strange and artificial, so we will try to motivate it with an example.

First, we choose a prime number. For simplicity, we will take ${p=5}$ . Normally, we express numbers as expansions in powers of 10, for example,

$\displaystyle 1024 = 1\times10^3 + 0\times10^2 + 2\times 10^1 + 4\times10^0$

with multipliers all in the range ${D_{10}=\{0, 1, 2, \dots , 9\}}$ . Fractions involve negative powers of 10:

$\displaystyle \textstyle{\frac{3}{8}} = 0.375 = 3\times 10^{-1} + 7\times 10^{-2} + 5\times 10^{-3} \,.$

In many cases, the decimal expansion is infinite:

$\displaystyle \textstyle{\frac{2}{3} } = 0.666\cdots = 6\times 10^{-1} + 6\times 10^{-2} + 6\times 10^{-3} + \dots$

but, with the Euclidean norm, the terms become ever-smaller.

Now we will change to base ${p=5}$ . By repeated division by ${5}$ , we easily find

$\displaystyle (1024)_{10} = (13044)_5 \,.$

What about fractions? We could try

$\displaystyle \textstyle{\frac{1}{2}} = \textstyle{\frac{1}{5-3}} = 5^{-1}( 1-\textstyle{\frac{3}{5}})^{-1} = 1\times 5^{-1} + 3\times 5^{-2} + 3^2\times 5^{-3} + 3^3\times 5^{-4} + \dots \,.$

But this is not in canonical form, because the `digits’ are not in the range ${D_5=\{0, 1, 2, 3, 4 \}}$ .

Let us try another approach: we seek to express ${\textstyle{\frac{1}{2}}}$ as a multiple ${d}$ of 5 and a remainder ${r}$ in ${D_5=\{0,1,2,3,4\}}$ :

$\displaystyle \textstyle{\frac{1}{2}} = d\times 5 + r$

Trying all ${r\in D_5}$ , we easily find that

$\displaystyle \textstyle{\frac{1}{2}} = (-\textstyle{\frac{1}{2}})\times 5 + 3 \,. \ \ \ \ \ (1)$

Now we do the same for ${(-\textstyle{\frac{1}{2}})}$ and find that

$\displaystyle (-\textstyle{\frac{1}{2}}) = (-\textstyle{\frac{1}{2}})\times 5 + 2 \,. \ \ \ \ \ (2)$

We can substitute (2) into (1) repeatedly to get

$\displaystyle \begin{array}{rcl} \textstyle{\frac{1}{2}} &=& (-\textstyle{\frac{1}{2}})\times 5 + 3 \hspace{5.0cm} = (-\textstyle{\frac{1}{2}})\times 5 + 3 \\ \null &=& ((-\textstyle{\frac{1}{2}})\times 5 + 3)\times 5 + 3 \hspace{2.0cm} = (-\textstyle{\frac{1}{2}})\times 5^2 + 2\times 5 + 3 \\ \null = && \hspace{-1.0cm} (((-\textstyle{\frac{1}{2}})\times 5 + 2)\times 5 + 3)\times 5 + 3 \hspace{0.0cm} = (-\textstyle{\frac{1}{2}})\times 5^3 + 2\times 5^2 + 2\times 5 + 3 \\ \end{array}$

For any ${n}$ , we then have

$\displaystyle \textstyle{\frac{1}{2}} = (-\textstyle{\frac{1}{2}})\times 5^n + (\underbrace{22\cdots2}_{(n-1)\ 2s}3)_5 \ \ \ \ \ (3)$

It appears that we are getting bigger numbers for increasing ${n}$ and, for the Euclidean norm, this is true. But we will introduce a new norm, for which ${5^n}$ gets smaller as ${n}$ increases. Thus, we can continue the process indefinitely, to get

$\displaystyle \textstyle{\frac{1}{2}} = ( \cdots 2 2 2 3)_5 \,, \ \ \ \ \ (4)$

where the dots on the left imply an infinite string of 2’s.

We can check that the arithmetic works for expansions like this. Add ${\cdots 2223_5}$ to itself, digit by digit, from right to left: ${3 + 3 = 6 = 1 + 1\cdot5}$ so we write 1 and carry one. For each other digit we have ${2+2+1 = 5 = 0 + 1\cdot5}$ so we write 0 and carry again. The result is ${ \cdots 0001}$ .

The p-adic Norm

We need a norm that makes ${5^n}$ small for large ${n}$ . For an integer m in $\mathbb{Z}$ , we define

$\displaystyle | m |_p = p^{-\rho} \,, \quad\mbox{where\ } \rho \mbox{\ is the largest power of\ } p \mbox{\ that divides}\ m.$

For rational numbers ${m/n}$ , we write ${m = r p^\rho}$ and ${n = s p^\sigma}$ , where ${r}$ and ${s}$ are both coprime with ${p}$ . Then

$\displaystyle \frac{m}{n} = \frac{r}{s}\times\frac{p^\rho}{p^\sigma}$

and we define the p-adic norm as

$\displaystyle \left| \frac{m}{n} \right|_p = \frac{p^{-\rho}}{p^{-\sigma}} = p^{\sigma-\rho} \,.$

This is a well-defined norm, albeit a strange-looking one. The size of a number in this norm is determined by the number of ${p}$ ‘s in its prime factorization. The more ${p}$ ‘s there are, the smaller the norm. Two numbers are close together if their difference is divisible by a high power of p. The higher the power, the closer the two numbers are together.

Looking back to (3), we see that the first right-hand term has p-adic norm

$\displaystyle \left| (-\textstyle{\frac{1}{2}})\times 5^n \right|_p = 5^{-n} \,.$

Clearly, this gets smaller with increasing ${n}$ , so it makes sense to continue the expansion indefinitely and write ${\frac{1}{2}}$ as the infinite expansion in (4).

Filling in the gaps

Now we define the field of p-adic numbers to be the completion of ${\mathbb{Q}}$ with respect to the p-adic norm. That is, we add to ${\mathbb{Q}}$ the limits of all sequences that are Cauchy in the p-adic norm. We denote the resulting complete field by ${\mathbb{Q}_p}$ , the field of p-adic numbers.

We hope to continue on the topic of p-adic numbers next week. HERE.

Sources

${\bullet}$ Gouvêa, Fernando Q., 2008: Local and Global Number Theory. § III.51 in The Princeton Companion to Mathematics, Ed. Timothy Gowers. Princeton Unversity Press (pp.~241-244). ISBN: 978-0-6911-1880-2.

${\bullet}$ Katok, Svetlana, 2007: P-adic Analysis Compared with Real. Amer. Math. Soc., 152pp. ISBN: 978-0-8218-4220-1.

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