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A division algebra, also called a "division ring" or "skew field," is a ring in which every nonzero element has a multiplicative inverse, but multiplication is not ...

The divergence of a vector field F, denoted div(F) or del ·F (the notation used in this work), is defined by a limit of the surface integral del ·F=lim_(V->0)(∮_SF·da)/V (1) ...

The definition of a set by enumerating its members. An extensional definition can always be reduced to an intentional one. An extension field is sometimes also called simply ...

A nonzero ring S whose only (two-sided) ideals are S itself and zero. Every commutative simple ring is a field. Every simple ring is a prime ring.

An additive group is a group where the operation is called addition and is denoted +. In an additive group, the identity element is called zero, and the inverse of the ...

There does not exist an everywhere nonzero tangent vector field on the 2-sphere S^2. This implies that somewhere on the surface of the Earth, there is a point with zero ...

A separable extension K of a field F is one in which every element's algebraic number minimal polynomial does not have multiple roots. In other words, the minimal polynomial ...

If a sequence takes only a small number of different values, then by regarding the values as the elements of a finite field, the Berlekamp-Massey algorithm is an efficient ...

A rational homomorphism phi:G->G^' defined over a field is called an isogeny when dimG=dimG^'. Two groups G and G^' are then called isogenous if there exists a third group ...

An algebraically soluble equation of odd prime degree which is irreducible in the natural field possesses either 1. Only a single real root, or 2. All real roots.