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...

{\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...

theorem Boolean prime ideal theorem Ideal theory Ideal (order theory) Ideal norm Splitting of prime ideals in Galois extensions Ideal sheaf Some authors...

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

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

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...

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...

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...

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

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

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...

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...

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...

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

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...

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

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...

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...

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

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

of xn − 1 are the powers of ζn, so Q(ζn) is the splitting field of xn − 1 (or of Φ(x)) over Q. Therefore Q(ζn) is a Galois extension of Q. The Galois...

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

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

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

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...

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...

The inverse Galois problem: is every finite group the Galois group of a Galois extension of the rationals? Are there an infinite number of Leinster groups...

This argument does not use Galois theory. However, Galois theory is required deduce from this the condition for the existence of the Jordan-Chevalley given...

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...