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Given a number field K, a Galois extension field L, and prime ideals p of K and P of L unramified over p, there exists a unique element sigma=((L/K),P) of the Galois group ...

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 ...

The German mathematician Kronecker proved that all the Galois extensions of the rationals Q with Abelian Galois groups are subfields of cyclotomic fields Q(mu_n), where mu_n ...

The Chebotarev density theorem is a complicated theorem in algebraic number theory which yields an asymptotic formula for the density of prime ideals of a number field K that ...

A solvable group is a group having a normal series such that each normal factor is Abelian. The special case of a solvable finite group is a group whose composition indices ...

An extension F of a field K is said to be algebraic if every element of F is algebraic over K (i.e., is the root of a nonzero polynomial with coefficients in K).

A separable algebraic extension E of F for which every irreducible polynomial in F which has a single root in E has all its roots in E is said to be Galoisian. Galoisian ...

The conjecture that the Artin L-function of any n-dimensional complex representation of the Galois group of a finite extension of the rational numbers Q is an Artin ...

A normal extension is the splitting field for a collection of polynomials. In the case of a finite algebraic extension, only one polynomial is necessary.

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 ...