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A totally imaginary field is a field with no real embeddings. A general number field K of degree n has s real embeddings (0<=s<=n) and 2t imaginary embeddings (0<=t<=n/2), ...

Given a number field K, there exists a unique maximal unramified Abelian extension L of K which contains all other unramified Abelian extensions of K. This finite field ...

A divergenceless field can be partitioned into a toroidal and a poloidal part. This separation is important in geo- and heliophysics, and in particular in dynamo theory and ...

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 ...

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, ...

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 ...