The p-Adic Numbers (Part 2)

Kurt Hensel (1861-1941)

Kurt Hensel, born in Königsberg, studied mathematics in Berlin and Bonn, under Kronecker and Weierstrass; Leopold Kronecker was his doctoral supervisor. In 1901, Hensel was appointed to a full professorship at the University of Marburg. He spent the rest of his career there, retiring in 1930.

Hensel is best known for his introduction of the p-adic numbers. They evoked little interest at first but later became increasingly important in number theory and other fields. Today, p-adics are considered by number theorists as being “just as good as the real numbers”. Hensel’s p-adics were first described in 1897, and much more completely in his books, Theorie der algebraischen Zahlen, published in 1908 and Zahlentheorie published in 1913.

The paper The p-Adic Numbers of Hensel (MacDuffee, 1938) opens as follows: “One cannot blame a respectable mathematician for looking twice at the equation

$\displaystyle -1 = 4 + 4\times5 + 4\times5^2 + 4\times5^3 + \cdots$

However, if we add 1 to both sides of this equation, we have 0’s as far out as we care to carry it.” The paper continues: “The whole point … is that this is not ordinary convergence, but a new type of convergence which, from the point of view of abstract algebra, is equally worthy of the name.”

The p-adic Numbers

In last week’s post (HERE), we examined how to fill the gaps in the rational number line by introducing additional numbers, the limits of sequences of rational numbers. This process of completion gives us the real numbers ${\mathbb{R}}$ .

A metric space ${\mathcal{M}}$ is a space that is endowed with a distance function ${d: \mathcal{M}\times\mathcal{M} \rightarrow \mathbb{R}}$ . With this function, we can define Cauchy sequences in ${\mathcal{M}}$ , sequences for which terms beyond some point are arbitrarily close (the definition is in the earlier post). A complete space is one that includes the limit of every Cauchy sequence. Every metric space can be completed: by adding to ${\mathcal{M}}$ the limits of all Cauchy sequences, we obtain the completion of ${\mathcal{M}}$ .

The reals are constructed using the usual Euclidean metric. We also found that there are other metrics that can be used to complete ${\mathbb{Q}}$ , which yield different number fields, one for every prime number ${p}$ . These are called the p-adic numbers, denoted ${\mathbb{Q}_p}$ . To construct the p-adic numbers, we define a new norm and a new measure of distance. We choose a specific prime number p. By the unique factorization theorem, any rational number ${q}$ can be expressed in the form

$\displaystyle q = (m/n) p^k$

where ${m}$ and ${n}$ are integers relatively prime to ${p}$ and the integer ${k}$ may be positive or negative. We then define the p-adic norm as follows:

$\displaystyle | q |_p = p^{-k} \,,$

that is, the inverse of the largest power of ${p}$ that divides ${q}$ . 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.

As an example, the number ${\frac{1001}{1000}}$ is close to ${1}$ in ${\mathbb{R}}$ , but it is quite distant in ${\mathbb{Q}_5}$ :

$\displaystyle \left| \frac{1001}{1000}-1 \right|_\infty = \left|0.001\right|_\infty = \frac{1}{1000} \,,\ \left| \frac{1001}{1000}-1 \right|_5 = \left|2^{-3}\times5^{-3}\right|_5 = 125$

The field of p-adic numbers is the completion of ${\mathbb{Q}}$ with respect to the p-adic norm.

The Triangle Inequality

The Euclidean norm, which we will write ${ |\ \ |_{\infty}}$ , satisfies the triangle inequality: for any real numbers ${x}$ and ${y}$ ,

$\displaystyle | x + y |_\infty \le | x |_\infty + | y |_\infty \,.$

In contrast, the p-adic norm satisfies what is called the strong triangle inequality:

$\displaystyle | x + y |_p \le \max \{ | x |_p, | y |_p \} \,.$

It allows us to introduce a so-called ultra-metric ${d_p(x,y) = |x-y|_p}$ , such that

$\displaystyle d_p(x, z) \le \max \{d_p(x,y) + d_p(y,z) \} \,.$

A strange consequence of this is that ${| n |_p\le 1}$ for every integer ${n}$ .

Ostrowski’s Theorem states that any (non-trivial) norm on ${\mathbb{Q}}$ is equivalent to either the Euclidean norm ${|\ \ |_\infty}$ or to a p-adic norm ${|\ \ |_p}$ for some prime number ${p}$ .

The usual Euclidean norm is Archimedean: given any two real numbers ${x}$ and ${y}$ , it is possible to find a finite number ${N}$ such that ${N|x|>|y|}$ ; enough small steps added together will exceed any large step. The ${p}$ -adic norm is non-Archimedean. There are numbers that cannot be compared in magnitude, and there is no order structure on ${\mathbb{Q}_p}$ .

There are numbers in ${\mathbb{Q}_p}$ that have no square root in ${\mathbb{Q}_p}$ . Using Hensel’s Lemma (Katok, 2007), we can easily show that ${\sqrt{6}}$ is in ${\mathbb{Q}_5}$ while ${\sqrt{7}}$ is not.

The p-adic Integers ${\mathbb{Z}_p}$

Any positive integer can be expanded in the form

$\displaystyle \sum_{k=0}^n a_k p^k \,,$

where ${a_k}$ is an integer in ${\{0, 1, \dots\ ,p-1\}}$ . Usually, we extend this to the rational and, ultimately, real numbers by extending the lower limit of the summation:

$\displaystyle \sum_{k=-\infty}^n a_k p^k \,,$

(adding digits to the right). For the p-adics, we extend the upper limit (adding digits to the left):

$\displaystyle \sum_{k=\ell}^{+\infty} a_k p^k \,,$

where ${\ell\in\mathbb{Z}}$ .

The p-adic numbers for which ${\ell\ge0}$ are called the p-adic integers and they form a sub-ring of the field ${\mathbb{Q}_p}$ , denoted by ${\mathbb{Z}_p}$ . We see that

$\displaystyle \mathbb{Z}_p = \{ x \in \mathbb{Q}_p : | x |_p \le 1 \} \,.$

The usual integers are in ${\mathbb{Z}_p}$ , but it also contains some surprises: since

$\displaystyle \frac{1}{2} = ( \dots 2223 )_5 = 3 + 2\times5 + 2\times5^2 + \cdots\ \,,$

we see that ${|\frac{1}{2}|_5 = 1}$ , so that ${\frac{1}{2}}$ is a 5-adic integer. Indeed, ${\mathbb{Z}_p}$ is uncountable, so it is `much larger’ than the set of integers ${\mathbb{Z}}$ . You may like to consider what other unexpected numbers are in ${\mathbb{Z}_5}$ . For example, do we have ${\sqrt{2}\in\mathbb{Z}_5}$ or ${\frac{22}{7}\in\mathbb{Z}_5}$ or ${\pi\in\mathbb{Z}_5}$ ?

Topology of ${\mathbb{Q}_p}$ compared to ${\mathbb{R}}$

In any metric space, we can define a topology generated by the set of open balls. Katok (2007) discusses the topology of the space ${(\mathbb{Q}_p,|\ \ |_p)}$ , and compares it to ${\mathbb{R}}$ with its usual topology, induced by the Euclidean metric. We note that the range of the p-adic metric is the discrete set ${\{0\}\cup\{p^k : k\in\mathbb{Z}\}}$ . This gives it topological properties that are rather strange. For example, ${\mathbb{Z}_p}$ has the topology of a Cantor ternary set.

For any open ball ${B(a,r) = \{ x : |x-a|_p\le r \}}$ in ${\mathbb{Q}_p}$ and any point ${b\in B(a,r)}$ in this ball, we have ${B(b,r)=B(a,r)}$ . Thus, every point in an open ball is the centre of the ball! Another curiousity is that in ultra-metric spaces, including ${(\mathbb{Q}_p,|\ \ |_p)}$ , every triangle is isosceles, with the equal sides longer than the third side. For more details, see Chapter 2 of Katok (2007).

The “Local-Global Principle”, formulated by Helmut Hasse, deserves another post. Perhaps when I understand it, I will return to this topic.

Sources

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

${\bullet}$ MacDuffee, C. C., 1938: The p-Adic Numbers of Hensel. Amer. Math. Monthly, 45, (8), 500–508.

${\bullet}$ Wikipedia article: p-adic Numbers. http://www.wikipedia.org/

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