P-adic analysis

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. 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 be the forward difference operator. Then for polynomial functions f we have the Newton series: where 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.

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