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For a field K with multiplicative identity 1, consider the numbers 2=1+1, 3=1+1+1, 4=1+1+1+1, etc. Either these numbers are all different, in which case we say that K has ...

The field axioms are generally written in additive and multiplicative pairs. name addition multiplication associativity (a+b)+c=a+(b+c) (ab)c=a(bc) commutativity a+b=b+a ...

A solenoidal vector field satisfies del ·B=0 (1) for every vector B, where del ·B is the divergence. If this condition is satisfied, there exists a vector A, known as the ...

The following are equivalent definitions for a Galois extension field (also simply known as a Galois extension) K of F. 1. K is the splitting field for a collection of ...

Let K be a number field of extension degree d over Q. Then an order O of K is a subring of the ring of integers of K with d generators over Z, including 1. The ring of ...

The degree (or relative degree, or index) of an extension field K/F, denoted [K:F], is the dimension of K as a vector space over F, i.e., [K:F]=dim_FK. If [K:F] is finite, ...

In particle physics, a spinor field of order 2s describes a particle of spin s, where s is an integer or half-integer. Therefore, a spinor of order 4s contains as much ...

The study of number fields by embedding them in a local field is called local class field theory. Information about an equation in a local field may give information about ...

The vector field N_f(z)=-(f(z))/(f^'(z)) arising in the definition of the Newtonian graph of a complex univariate polynomial f (Smale 1985, Shub et al. 1988, Kozen and ...

The following conditions are equivalent for a conservative vector field on a particular domain D: 1. For any oriented simple closed curve C, the line integral ∮_CF·ds=0. 2. ...