I have repeatedly bumped up against p-adic numbers and p-adic analysis in my reading and have wanted to delve more deeply into them for ages. In this first (rather long) note about them I wanted to explore some of the details involved in the close analogy between
A). the way Cantor constructed the real number field as the completion of the rationals using Cauchy sequences with the usual Euclidean metric, and
B). the way the p-adic number field can be similarly constructed as the completion of the rationals, but using Cauchy sequences with a different metric (known as an ultrametric).
I have found that exploring this analogy in some detail has allowed me to get quite a good foothold on some of the key features of p-adic analysis.
1. A basic initial characterisation of p-adic numbers
A lot flows from the basic observation that given a prime number and a rational number , it is always possible to factor out the powers of in as in the equation
The exponent , known as the p-adic valuation in the literature, can be negative, zero or positive depending on how the prime appears (or not) as a factor in the numerator and denominator of the rational number .
Example: Suppose we specify the prime . Then we can factor out the powers of 7 in the rational numbers 56/12, 177553, and 3/686 as
For each prime number , we can write any positive rational number in the power series form
where is the p-adic valuation of and the coefficients come from the set of least positive residues of . (These coefficients will always exhibit a repeating pattern in the power series of a rational number). This power series form is called the p-adic expansion of . In the case , i.e., when is a positive integer, the p-adic expansion is just the expansion of in base .
If the rational number is negative rather than positive, its p-adic expansion can be obtained from the positive version above as
where and for .
We can obtain the p-adic expansion for any rational number by the following algorithm. Let the p-adic expansion we want to find be
where all fractions are always given in their lowest terms. We deduce that the p-adic expansion of is then
so, since the right hand side equals mod , we can compute as
Next, we see that
We deduce that the p-adic expansion of is then
so, since the right hand side equals mod , we can compute as
We continue this process until the repeating pattern in the coefficients is spotted.
Example: Suppose we specify the prime to be and consider the rational number 2/15. The p-adic valuation of this rational number is -1 since we can write
Therefore we expect the p-adic expansion for 2/15 to have in its first term. Following the steps of the algorithm we compute the first coefficient as
Then we have
so we can compute the second coefficient as
Then we have
so we can compute the third coefficient as
Then we have
so we see immediately that the fourth coefficient will be the same as the second, and the pattern will repeat from this point onwards. Therefore we have obtained the p-adic expansion of 2/15 as
It can be shown that the set of all p-adic expansions is an algebraic field. This is called the field of p-adic numbers and is usually denoted by in the literature. In the rest of this note I will explore some aspects of the construction of the field by analogy with the way Cantor constructed the field of real numbers from the field of rationals. The next section reviews Cantor’s construction of the reals.
2. Cantor’s construction of the real number field
In Cantor’s construction of the real numbers from the rationals, we regard the latter as a metric space where the metric is defined in terms of the ordinary Euclidean absolute value function:
The central problem in constructing the real number field from the field of rationals is that of defining irrational numbers only in terms of rationals. This can be done in alternative ways, e.g., using Dedekind cuts, but Cantor’s approach achieves it using the concept of a Cauchy sequence. A Cauchy sequence in a metric space is a sequence of points which become arbitrarily ‘close’ to each other with respect to the metric, as we move further and further out in the sequence. More formally, in the context of the metric space , a sequence of rationals is a Cauchy sequence if for each (where ) there is an such that
Example: So, for example, if I set , then I can be sure that there is a certain point in the Cauchy sequence beyond which all the terms of the sequence will always be within a millionth of each other in absolute value. If instead I set I might have to go further out in the sequence, but I can still be sure that beyond a certain point all the terms of the sequence from then on will be within a billionth of each other in absolute value. And so on. END
Cantor’s approach to constructing the reals is based on the idea that any irrational number can be regarded as the limit of Cauchy sequences of rationals, so we can actually define the irrationals as sets of Cauchy sequences of rationals.
Example: To illustrate, consider the irrational number . Define three sequences of rationals , , as follows:
if , otherwise
if , otherwise
For each , lies between and , and at each iteration the closed interval has length
(To see this, note that from the definitions of the three sequences above we find that
so we get the result from this by a simple induction). Also, each closed interval contains . Therefore the closed interval is increasingly ‘closing in’ around , i.e., we have
So for each of the sequences , , of rationals, the terms of the sequence are getting closer and closer to each other, and closer to . Cantor’s idea was basically to define an irrational such as to be the set containing all Cauchy sequences like , , and which converge to that irrational. END
Formally, the process involves defining equivalence classes of Cauchy sequences in the metric space , so that two Cauchy sequences and belong to the same equivalence class, denoted , if for each (where ) there is an such that
It is straightforward to show that is an equivalence relation in the sense that it is reflexive (i.e. for all Cauchy sequences ), symmetric (i.e., if then for all Cauchy sequences and ), and transitive (i.e., if and then for all Cauchy sequences , and ).
Cantor defined a real number to be any equivalence class arising from , i.e., any set of the form
where is a Cauchy sequence in the metric space . Rational numbers are, of course, subsumed in this since any rational number belongs to the (constant) Cauchy sequence defined by for all .
It is now possible to define all the standard relations and arithmetic operations on the real numbers constructed in this way, and it can also be shown that the set of reals constructed in this way is isomorphic to the set of reals defined by alternative means, such as Dedekind cuts.
The set of reals constructed in this way can be regarded as the completion of the set of rationals in the sense that it is obtained by adding to the set of rationals all the limits of all possible Cauchy sequences in which are irrational. In general, a metric space is said to be complete if every Cauchy sequence in that metric space converges to a point within that metric space. Clearly, therefore, the metric space is not complete since, for example, we found Cauchy sequences of rationals above which converge to . However, it is a basic result of elementary real analysis that the metric space is complete. It is also a basic result that the completion of a field gives another field, so since is a field it must also be the case that is a field.
3. Archimedian vs. non-archimedian absolute values and ultrametric spaces
In constructing the p-adic number field it becomes important to distinguish between two types of absolute value function on a field, namely archimedian and non-archimedian absolute values. All absolute values on a field by definition have the properties that they assign the value 0 only to the field element 0, they assign the positive value to each non-zero field element , and they satisfy and the usual triangle inequality
The usual Euclidean absolute value function used in (1) above on , of course, satisfies these conditions, and is called archimedian because it has the property that there is no limit to the size of the absolute values that can be assigned to integers. We can write this as
Non-archimedian absolute values do not have this property. In addition to the basic conditions that all absolute values must satisfy, non-archimedian absolute values must also satisfy the additional condition
which is known as the ultrametric triangle inequality. It is obviously the case that the archimedian absolute value function used in (1) does not satisfy (2), e.g.,
In fact it follows from (2) that non-archimedian absolute values of integers can never exceed 1, because if (2) is to hold for the absolute value function then we can write for any integer n:
and so by induction we must have
Then if we would have which implies , and so in this case. It is not possible for to exceed 1, so in the case of non-archimedian absolute values we have
Any absolute value function which does not satisfy (2) is called archimedian, and these are the only two possible types of absolute values. To see that these are the only two possible types, suppose we have an absolute value function such that
where . Then there must exist an integer whose absolute value exceeds 1, and so gets arbitrarily large as k grows, so cannot be finite. The absolute value function must be archimedian in this case. Otherwise, we must have , but since for all absolute values it must be the case that , it must be the case that if is finite. Thus we must have a non-archimedian absolute value in this case and there are no other possibilities.
The trick in constructing the p-adic number field from the rationals is to use a certain non-archimedian absolute value function satisfying (2) to define the metric over , rather than the usual archimedian absolute value function in (1). In this regard we have the following:
Theorem 1. Define a metric on a field by . Then the absolute value function in this definition is non-archimedian if and only if for all field elements , , we have
Proof: Suppose first that the absolute value function is non-archimedian. Applying it to the equation
Conversely, suppose the given metric inequality holds. Then setting and in the metric inequality we get
which is equivalent to
thus proving that the absolute value function is non-archimedian. QED
A metric for which the inequality in Theorem 1 is true is called an ultrametric, and a space endowed with an ultrametric is called an ultrametric space. Such spaces have curious properties which have been studied extensively. In some ways, however, using a non-archimedian absolute value makes analysis much easier than in the usual archimedian case. In this regard we have the following result pertaining to Cauchy sequences with respect to a non-archimedian absolute value function, which is NOT true for archimedian absolute values:
Theorem 2. A sequence of rational numbers is a Cauchy sequence with respect to a non-archimedian absolute value if and only if we have
Proof: Letting , we have
because the absolute value is non-archimedian. We then have that if is Cauchy then the terms get arbitrarily closer as so we must have . Conversely, if is true, then we must also have for any , so the conditions of a Cauchy sequence are satisfied. QED
It is important to note that Theorem 2 is false for archimedian absolute values. The classic counterexample involves the partial sums of the harmonic series (which is divergent in terms of Euclidean absolute values). Consider the following three partial sums in particular:
Then we have
so the condition of Theorem 2 is satisfied. However,
Therefore it is not true in this case that for any , so the conditions of a Cauchy sequence are not satisfied here. It’s only in the context of non-archimedian absolute values that this works.
4. Constructing the p-adic number field as the completion of the rationals
To obtain the p-adic number field as the completion of the field of rationals in a way analogous to how Cantor obtained the reals from the rationals, we use an ultrametric based on a non-archimedian absolute value known as the p-adic absolute value.
For each prime there is an associated p-adic absolute value on obtained by factoring out the powers of in any given rational to get
With this factorisation in hand, the p-adic absolute value of is then defined as
if , and we set . (As mentioned earlier, the number is called the p-adic valuation of ).
It is straightforward to verify that this is a non-archimedian absolute value on . It has some surprising features. For example, unlike the usual Euclidean absolute value function on which can take any non-negative value on a continuum, the p-adic absolute value function can only take values in the discrete set
Examples: In the case of the 7-adic absolute value we have
Note that the 7-adic absolute value of 3/686 is large, while that of 35 is small. END
Now consider the metric space where is defined as
By virtue of Theorem 1, is an ultrametric and is an ultrametric space. Since the p-adic absolute value function has some counterintuitive features, it is not surprising that also gives some counterintuitive results. For example, the numbers 1627 and 2 are much ‘closer’ to each other with regard to this ultrametric than the numbers 3 and 2, because
In addition, we can use it to show that the sequence where
is Cauchy with respect to , whereas it is violently non-Cauchy with respect to the usual Euclidean absolute value. We have
It follows from Theorem 2 in the previous section that the sequence is Cauchy with respect to the p-adic absolute value. In fact, the infinite series
has the sum in the ultrametric space (this formula can be derived in the usual way for geometric series) but its sum is undefined in .
The following Theorem proves that the ultrametic space is not complete in a way which is analogous to how is not complete.
Theorem 3. The field of rational numbers is not complete with respect to the p-adic absolute value.
Proof: To prove this, we will create a Cauchy sequence with respect to the p-adic absolute value function whose limit does not belong to .
Let be an integer. Recall that a property of the Euler totient function is that for any prime and any integer we have
Also recall the Euler-Fermat Theorem which says that if then
With these in hand, consider the sequence . Then since we have , the Euler-Fermat Theorem tells us that
must be divisible by , so we have
and so the sequence is Cauchy with respect to the p-adic absolute value, by virtue of Theorem 2. If we call the limit of this sequence
we can write the following:
Therefore since , the limit of the sequence must be a nontrivial -th root of unity, so it cannot belong to . This proves that the ultrametric space is not complete. QED
Although is not complete with regard to the p-adic absolute value, we can construct the p-adic completion in a manner analogous to Cantor’s construction of as a completion of . Investigating the fine details of this and the properties of then lead one into the rich literature on p-adic analysis, which I will explore in future notes.