The $p$-adics and Related Concepts
Recall how we construct the real numbers from $\mathbb Q$. Here $| \cdot |$ is the usual absolute value, but the whole point of this section is that this construction generalizes for other absolute values. The basic idea is that we may make (new) real numbers by including limits of convergent sequences of rational numbers.
Definition: A sequence of rational numbers $(a_n)$ is called Cauchy if given $\epsilon$ there exists $N$ such that $n, m > N$ implies $|a_n – a_m| < \epsilon.$
Cauchy sequences are those for which the elements in the sequence are getting closer and closer together. You may remember Cauchy sequences from the theorem that says Cauchy sequences are convergent (and vice-versa). The important observation is that we can determine whether or not a sequence is convergent by only referring to elements of the sequence.
Definition: Two sequences of rational numbers $(a_n)$ and $(b_n)$ are equivalent if $(a_n – b_n) \rightarrow 0$. That is if given $\epsilon$ there exists $N$ such that $n > N$ implies $|a_n – b_n | < \epsilon$. We write $[a_n]$ for the equivalence class of $(a_n)$.
Definition: The set of equivalence classes $\{ [a_n] : (a_n) \subset \mathbb Q \}$ is called the Cauchy completion of $\mathbb Q$ with respect to $| \cdot |$. When this is the usual absolute value then the Cauchy completion of $\mathbb Q$ is the real numbers.
You probably don’t think about the real numbers as equivalence classes of Cauchy sequences. To the extent you think of generic real numbers at all it is probably in terms of its decimal expansion. However, as there are no patterns among the digits of a generic real number, if you represent one as, say, $\pi = 3.14159 \cdots$, you are actually providing a rational approximation to $\pi$ valid out to however many decimal places you report. Note that this is equivalent to giving the beginnings of a Cauchy sequence $(3, 3.1, 3.14, 3.141, 3.1415, 3.14159, \ldots)$ and so in fact the notion of decimal expansion of real numbers can be explained in terms of Cauchy sequences. The fact that there are multiple representations for a given real number is likely not new to you either. For instance, if you expand a real numbers in base $b \neq 10$ then we get a different representation than the standard decimal expansion. For instance if $x = d_0. d_1 d_2 d_3 \cdots$ base $b$, then $$x = d_0 + d_1 b^{-1} + d_2 b^{-2} + d_3 b^{-3} + \cdots.$$ is a Cauchy sequence for $x$ which is in the same equivalence class as that given by the decimal expansion of $x$. Perhaps the most natural reason that real numbers are naturally defined as equivalence classes is, if our original notion of real numbers was to include all limits of convergent sequences, then each limit is the limit of lots of different sequences—the decimal expansion gives just one of these.
How do we get the arithmetic operations on the real numbers if we view them as equivalence classes of Cauchy sequences? The answer is the limit laws, which we can phrase purely in terms of Cauchy sequences.
Definition: The arithmetic operations $+, -, \cdot, /$ extend to equivalence classes of Cauchy sequences. That is, if $[a_n]$ and $[b_n]$ are equivalence classes of Cauchy sequences represented by $(a_n)$ and $(b_n)$, then
- $[a_n] + [b_n] := [a_n + b_n]$
- $[a_n] – [b_n] := [a_n – b_n]$
- $[a_n] \cdot [b_n] := [a_n \cdot b_n]$
- If $[b_n] \neq [0]$ then we can find equivalent $(b_n’)$ with non-zero entries and $[a_n]/[b_n] := [a_n/b_n].$
This definition requires a proof—it is not immediate that these arithmetic operations are well-defined.
Exercise: Show that the algebraic operations on $\mathbb R$ are well-defined.
The $p$-adic Rational Numbers
Nothing in the construction of the real numbers was specialized to $| \cdot |_{\infty}$ and the $p$-adic rational numbers result when we use $| \cdot |_p$ in the Cauchy completion instead.
Definition: The Cauchy completion of $\mathbb Q$ with respect to $| \cdot |_p$ is called the $p$-adic rational numbers, denoted $\mathbb Q_p$. It is a field.
In spite of the similarity in construction, the $p$-adic rational numbers has some striking differences with $\mathbb R$. One immediate observation, is that if $(m_n)$ is a sequence of integers, then $(m_n)$ is Cauchy with respect to $| \cdot |_{\infty}$ if and only if $(m_n)$ is eventually constant. That is $[(m_n)] = [(M)]$ for some $M \in \mathbb Z$. It follows that the Cauchy completion of $\mathbb Z$ with respect to $| \cdot |_{\infty}$ produces no new real numbers.
On the other hand, if $p$ is a finite prime, then $| m_n |_p \leq 1$ and hence it is possible that a sequence of integers converges to a non-integer $p$-adic rational number. It turns out that the Cauchy completion of $\mathbb Z$ with respect to $| \cdot |_p$ introduces new $p$-adic numbers. We call this Cauchy completion of $\mathbb Z$ the $p$-adic integers and denote it $\mathbb Z_p$.
We conclude by noting that we have discovered all Cauchy completions of $\mathbb Q$. This is captured in the following trio of theorems.
Theorem: The Cauchy completion of $\mathbb Q$ with respect to the trivial absolute value is $\mathbb Q$.
Theorem: If $| \cdot |$ and $\| \cdot \|$ are equivalent absolute values on $\mathbb Q$, then their Cauchy completions are the same.
Theorem: If $| \cdot |_p$ and $| \cdot |_q$ are absolute values in different places of $\mathbb Q$, then there exists a sequence of rational numbers $(a_n)$ which is Cauchy with respect to $| \cdot |_p$ but not Cauchy with respect to $| \cdot |_q$. That is $\mathbb Q_p \neq \mathbb Q_q$.