The main purposes of this paper are (i) to establish weighted-semicircular elements in a certain Banach *-probability space induced by measurable function s on

p-adic number fields [??], for primes p, (ii) to consider free-distributional data of our weighted-semicircular elements, and those of operators generated by these elements under free product, (iii) to investigate semicircular elements generated by the weighted-semicircular elements of (ii), and (iv) to study how [??] acts on our weighted-semicircular, or corresponding semicircular elements, in the sense of free stochastic calculus in the sense of [29].

Rim, "Generalized Carlitz's Euler Numbers in the

p-adic number field," Advanced Studies in Contemporary Mathematics, vol.

The completion of Q with respect to | | is denoted by [Q.sub.p] and is called the

p-adic number field.

When one talks of q-extension, q is considered in many ways such as an indeterminate, a complex number q [member of] C, or

p-adic number q [member of] [C.sub.p].

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.

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.