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In an additive group G, the additive inverse of an element a is the element a^' such that a+a^'=a^'+a=0, where 0 is the additive identity of G. Usually, the additive inverse ...

An Artin L-function over the rationals Q encodes in a generating function information about how an irreducible monic polynomial over Z factors when reduced modulo each prime. ...

A sequence a_1, a_2, ... such that the metric d(a_m,a_n) satisfies lim_(min(m,n)->infty)d(a_m,a_n)=0. Cauchy sequences in the rationals do not necessarily converge, but they ...

A group having continuous group operations. A continuous group is necessarily infinite, since an infinite group just has to contain an infinite number of elements. But some ...

A basis for the real numbers R, considered as a vector space over the rationals Q, i.e., a set of real numbers {U_alpha} such that every real number beta has a unique ...

A collection of equations satisfies the Hasse principle if, whenever one of the equations has solutions in R and all the Q_p, then the equations have solutions in the ...

Let K be a field of field characteristic 0 (e.g., the rationals Q) and let {u_n} be a sequence of elements of K which satisfies a difference equation of the form ...

The winding number W(theta) of a map f(theta) with initial value theta is defined by W(theta)=lim_(n->infty)(f^n(theta)-theta)/n, which represents the average increase in the ...

A perfect field is a field F such that every algebraic extension is separable. Any field in field characteristic zero, such as the rationals or the p-adics, or any finite ...

A condition used in the definition of a mathematical object, commonly denoted : or |. For example, the rationals Q can be defined by Q={p/q:q!=0,p,q in Z}, read as "the set ...