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A p-adic number is an extension of the field of rationals such that congruences modulo powers of a fixed prime p are related to proximity in the so called "p-adic metric." ...

In simple terms, let x, y, and z be members of an algebra. Then the algebra is said to be associative if x·(y·z)=(x·y)·z, (1) where · denotes multiplication. More formally, ...

The German mathematician Kronecker proved that all the Galois extensions of the rationals Q with Abelian Galois groups are subfields of cyclotomic fields Q(mu_n), where mu_n ...

A general reciprocity theorem for all orders which covered all other known reciprocity theorems when proved by E. Artin in 1927. If R is a number field and R^' a finite ...

The group of functions from an object G to itself which preserve the structure of the object, denoted Aut(G). The automorphism group of a group preserves the multiplication ...

An experiment E(S,F,P) is defined (Papoulis 1984, p. 30) as a mathematical object consisting of the following elements. 1. A set S (the probability space) of elements. 2. A ...

A polynomial admitting a multiplicative inverse. In the polynomial ring R[x], where R is an integral domain, the invertible polynomials are precisely the constant polynomials ...

The identity element of an additive group G, usually denoted 0. In the additive group of vectors, the additive identity is the zero vector 0, in the additive group of ...

Let A be a matrix and x and b vectors. Then the system Ax=b, x>=0 has no solution iff the system A^(T)y>=0, b^(T)y<0 has a solution, where y is a vector (Fang and Puthenpura ...

An algebraic ring which appears in treatments of duality in algebraic geometry. Let A be a local Artinian ring with m subset A its maximal ideal. Then A is a Gorenstein ring ...