F. Gouvêa's "$p$-adic Numbers. An Introduction" motivates the titular object in mainly two different ways. First, there is the idea of succesively lifting solutions modulo prime powers formalised by Hensel's Lemma and, second, there is something he dubs Hensel's Analogy. Let me expand on the latter (which is more or less the table on page $1$ here).
Recall that any positive integer $n$ can be expanded into powers of any fixed prime number $p$ defining its $p$-adic expansion. More explicitly, there are non-negative integers $a_k$ for $k=0,\dots,m$ so that $0\le a_i<p$ and
$$
n=\sum_{k=0}^m a_kp^k\,.
$$
Put differently, any positive integer is expressible as a polynomial in $p$ with coefficients between $0$ and $p-1$. It then seems natural to ask whether this can be extended to all (positive) rational numbers by somewhat simplifying the resulting fraction of polynomials.
Drawing from complex analysis we know that for polynomials $P(z),Q(z)\in\mathbb C[z]$ we can expand
$$
\frac{P(z)}{Q(z)}
=\sum_{k=-m}^\infty a_k(z-\alpha)^k
$$
for, say, $\alpha$ a root of $Q(z)$. This is the Laurent series of the rational function. Jumping ahead, the $p$-adic numbers $\mathbb Q_p$ give a way of making sense of such infinite series in powers of the fixed prime $p$ corresponding to the "local" study at $p$. The crucial technical part here is to introduce a sensible notion of convergence to allow infinite power series in place of mere polynomials. We arrive then at $\mathbb Q_p$ having as elements
$$
\sum_{k=-m}^\infty a_kp^k
$$
with integers $0\le a_k<p$. This naturally extends the $p$-adic expansions we are used to while simultaneously introducing a very useful and powerful number theoretic tool.
However, I was wondering: Is there more to Hensel's Analogy then formal similarities? Conceptually, studying the local fields $\mathbb Q_p$ is a way of focusing on the prime $p$. For example, any number field $K/\mathbb Q$ has a corresponding local extensions $K_{\mathfrak p}/\mathbb Q_p$ with $\mathfrak p\mid p$ in which we can more easily study the, say, ramification behavior of $K/\mathbb Q$ at $p$. So the $p$-adic numbers provide a framework for the "local" study near $p$ similar to what a Laurent series expansion does for a rational function (which is also an idea close to algebraic geometry). Hence the analogy seems to hold some truth.
But how much truth does Hensel's Analogy really hold? That is, what non-trivial techniques or results (if any) can be carried over from complex analysis to number theory? What can be said besides the things which follow from the formal equity to the Laurent series?
Thanks in advance!
Best Answer
In complex analysis, an analytic function on a closed disc that is not identically zero has finitely many zeros in the disc due to compactness of the disc and the fact that zeros of an analytic function of a single complex variable are isolated (if the function is not identically $0$). This carries over to $p$-adic analytic functions on $\mathbf Z_p$, which contains $\mathbf Z$, and that suggests the possibility of proving a Diophantine equation has finitely many solutions in $\mathbf Z$ by proving it has finitely many solutions in $\mathbf Z_p$ for a suitable prime $p$ by placing the integral solutions among the zeros of a $p$-adic analytic function. This idea leads to Skolem's method, for example.
For another example, the last section here shows an analogy between Rouché's theorem in complex analysis and Strassmann's theorem in $p$-adic analysis.