In mathematics, p-adic analysis is a branch of number theory that deals with the mathematical analysis of functions of p-adic numbers.

The theory of complex-valued numerical functions on the p-adic numbers is part of the theory of locally compact groups. The usual meaning taken for p-adic analysis is the theory of p-adic-valued functions on spaces of interest.

Applications of p-adic analysis have mainly been in number theory, where it has a significant role in diophantine geometry and diophantine approximation. Some applications have required the development of p-adic functional analysis and spectral theory. In many ways p-adic analysis is less subtle than classical analysis, since the ultrametric inequality means, for example, that convergence of infinite series of p-adic numbers is much simpler. Topological vector spaces over p-adic fields show distinctive features; for example aspects relating to convexity and the Hahn–Banach theorem are different.

## Important results

### Ostrowski's theorem

Ostrowski's theorem, due to Alexander Ostrowski (1916), states that every non-trivial absolute value on the rational numbers Q is equivalent to either the usual real absolute value or a p-adic absolute value. 

### Mahler's theorem

Mahler's theorem, introduced by Kurt Mahler,  expresses continuous p-adic functions in terms of polynomials.

In any field of characteristic 0, one has the following result. Let

$(\Delta f)(x)=f(x+1)-f(x)$ be the forward difference operator. Then for polynomial functions f we have the Newton series:

$f(x)=\sum _{k=0}^{\infty }(\Delta ^{k}f)(0){x \choose k},$ where

${x \choose k}={\frac {x(x-1)(x-2)\cdots (x-k+1)}{k!}}$ is the kth binomial coefficient polynomial.

Over the field of real numbers, the assumption that the function f is a polynomial can be weakened, but it cannot be weakened all the way down to mere continuity.

Mahler proved the following result:

Mahler's theorem: If f is a continuous p-adic-valued function on the p-adic integers then the same identity holds.

### Hensel's lemma

Hensel's lemma, also known as Hensel's lifting lemma, named after Kurt Hensel, is a result in modular arithmetic, stating that if a polynomial equation has a simple root modulo a prime number p, then this root corresponds to a unique root of the same equation modulo any higher power of p, which can be found by iteratively " lifting" the solution modulo successive powers of p. More generally it is used as a generic name for analogues for complete commutative rings (including p-adic fields in particular) of the Newton method for solving equations. Since p-adic analysis is in some ways simpler than real analysis, there are relatively easy criteria guaranteeing a root of a polynomial.

To state the result, let $f(x)$ be a polynomial with integer (or p-adic integer) coefficients, and let m,k be positive integers such that mk. If r is an integer such that

$f(r)\equiv 0{\pmod {p^{k}}}$ and $f'(r)\not \equiv 0{\pmod {p}}$ then there exists an integer s such that

$f(s)\equiv 0{\pmod {p^{k+m}}}$ and $r\equiv s{\pmod {p^{k}}}.$ Furthermore, this s is unique modulo pk+m, and can be computed explicitly as

$s=r+tp^{k}$ where $t=-{\frac {f(r)}{p^{k}}}\cdot (f'(r)^{-1}).$ 