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A finite field is a field with a finite field order (i.e., number of elements), also called a Galois field. The order of a finite field is always a prime or a power of a ...

A complete set of mutually conjugate group elements. Each element in a group belongs to exactly one class, and the identity element (I=1) is always in its own class. The ...

A field automorphism of a field F is a bijective map sigma:F->F that preserves all of F's algebraic properties, more precisely, it is an isomorphism. For example, complex ...

Given a field F and an extension field K superset= F, if alpha in K is an algebraic element over F, the minimal polynomial of alpha over F is the unique monic irreducible ...

The extension field K of a field F is called a splitting field for the polynomial f(x) in F[x] if f(x) factors completely into linear factors in K[x] and f(x) does not factor ...

The ith Pontryagin class of a vector bundle is (-1)^i times the ith Chern class of the complexification of the vector bundle. It is also in the 4ith cohomology group of the ...

When working over a collection of fields, the base field is the intersection of the fields in the collection, i.e., the field contained in all other fields.

Let V be a vector space over a field K, and let A be a nonempty set. For an appropriately defined affine space A, K is called the coefficient field.

The ith Stiefel-Whitney class of a real vector bundle (or tangent bundle or a real manifold) is in the ith cohomology group of the base space involved. It is an obstruction ...

An imaginary quadratic field is a quadratic field Q(sqrt(D)) with D<0. Special cases are summarized in the following table. D field members -1 Gaussian integer -3 Eisenstein ...