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 [T.sub.v]([p.sup.l]) satisfy

Other topics are rational points on elliptic curves, conics and the

p-adic numbers, the zeta function, and algebraic number theory.

p-Adic Numbers, 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.