Koenigsmann has obtained many results on Grothendieck's program of anabelian geometry as well: for example, the first proof of the birational variant of the section conjecture for all smooth projective curves of genus >1 over the field of p-adic numbers
is due to him, as well as the disproof of the conjecture and of its birational variant for the field of p-adic algebraic numbers.
He does assume they have already seen some non-archimedean fields, usually the p-adic numbers
and hopefully the complete algebraically closed p-adic field Cp, but he reviews them briefly just in case.
Fast Iterative Division of P-adic Numbers
. IEEE Transactions on Computers.
Lemma 2.11 For v [greater than or equal to] -1, the p-adic numbers
Other topics are rational points on elliptic curves, conics and the p-adic numbers
, the zeta function, and algebraic number theory.
, Ultrametric Analysis and Applications, Vol.
In 1897, Hensel (5) discovered the p-adic numbers as a number theoretical analogue of power series in complex analysis.
most important examples of non-Archimedean spaces are p-adic numbers. A key property of p-adic numbers is that they do not satisfy the Archimedean axiom: for all x and y > 0, there exists an integer n such that x < ny:
Algebra moved from more concrete concerns in solving equations and finding regularities within number theory to maneuvers of inventing new types of numbers for various tasks (ideal numbers, quaternions, p-adic numbers
Using mainly concrete constructions, Gerstein gives a brief introduction to classical forms, then moves to quadratic spaces and lattices, valuations, local fields, p-adic numbers
, quadratic spaces over Qp and over Q, lattices over principal ideal domains, initial integral results, the local-global approach to lattices, and applications to cryptography.
They conclude by explaining the field of p-adic numbers
, their squares, absolute values and valuations, the topologies of valuation type, local fields and locally compact fields.