products of prime ideals of OL, provides one of the richest parts of algebraic number theory. The splitting of prime ideals in Galois extensions is sometimes...

Ideal theory Ideal (order theory) Ideal norm Splitting of prime ideals in Galois extensions Ideal sheaf Some authors call the zero and unit ideals of...

field can be split (factored) in a Galois extension. See Covering map and Splitting of prime ideals in Galois extensions for more details. Closed geodesics...

Chebotarev density theorem in algebraic number theory describes statistically the splitting of primes in a given Galois extension K of the field Q {\displaystyle...

In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any...

{\mathcal {O}}_{K}} of a quadratic field K {\displaystyle K} . In line with general theory of splitting of prime ideals in Galois extensions, this may be p...

separable extension K′ of K, a Galois closure L of K′ is a type of splitting field, and also a Galois extension of K containing K′ that is minimal, in an obvious...

Splitting of prime ideals in Galois extensions. The same idea in the proof shows that if L / K {\displaystyle L/K} is a purely inseparable extension (need...

Heegner number Langlands program Different ideal Dedekind domain Splitting of prime ideals in Galois extensions Decomposition group Inertia group Frobenius...

integer Splitting of prime ideals in Galois extensions describes the structure of prime ideals in the Gaussian integers Table of Gaussian integer factorizations...

Sea: Foundations of algebraic geometry (PDF). Retrieved 5 June 2019. "Splitting and ramification in number fields and Galois extensions". PlanetMath....

Q by the roots of 2x5 − 32x + 1, which has Galois group S5. Splitting of prime ideals in Galois extensions Narkiewicz (1990) p.416 Narkiewicz (1990) p...

differential Galois theory, a variant of Galois theory dealing with linear differential equations. Galois theory studies algebraic extensions of a field by...

fundamental theorem of Galois theory. Field extensions can be generalized to ring extensions which consist of a ring and one of its subrings. A closer...

the extension field in which a polynomial can be factored into its roots is known as the splitting field of the polynomial. The Galois group of a polynomial...

structure of the set of extensions is known better when L/K is Galois. Let (K, v) be a valued field and let L be a finite Galois extension of K. Let Sv...

reciprocity laws as a statement that the Artin symbol from ideals (or ideles) to elements of a Galois group is trivial on a certain subgroup. Several more recent...

is a splitting field which is a separable extension of K of degree equal to the index of A, and this splitting field is isomorphic to a subfield of A. As...

exploited in Grothendieck's Galois theory). It can be shown that for separable extensions the radical is always {0}; therefore the Galois theory case...

describing the solution of the linear least squares problem Normal extensions (or quasi-Galois), field extensions, splitting fields for a set of polynomials over...

(up to reordering) as a product of primes. Galois A Galois extension is a finite field extension L/K such that one of the following equivalent conditions...

intersections of, respectively, all algebraic ideals, all differential ideals, and all radical differential ideals that contain it. The algebraic ideal generated...

field extension. In general, the absolute Galois group of K is the Galois group of Ksep over K. Algebraically closed field Algebraic extension Puiseux...

provide fresh information on the splitting of prime ideals in a Galois extension; a common way to explain the objective of a non-abelian class field theory...

distinct. The inverse Galois problem: is every finite group the Galois group of a Galois extension of the rationals? Isomorphism problem of Coxeter groups Are...

Like many other finite groups, it can be realized as the Galois group of a certain field of algebraic numbers. The quaternion group Q8 has the same order...

{\displaystyle \mathbb {Q} (\zeta _{n})} is a Galois extension of Q {\displaystyle \mathbb {Q} } . The Galois group Gal ⁡ ( Q ( ζ n ) / Q ) {\displaystyle...

Noether determined the minimal set of conditions required that a primary ideal be representable as a power of prime ideals, as Richard Dedekind had done for...

In mathematics, the norm residue isomorphism theorem is a long-sought result relating Milnor K-theory and Galois cohomology. The result has a relatively...

function Formally real field Real closed field Applications Galois theory Galois group Inverse Galois problem Kummer theory General Module (mathematics) Bimodule...