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Splitting of prime ideals in Galois extensions

In mathematics, the interplay between the Galois group G of a Galois extension L of a number field K, and the way the prime ideals P of the ring of integers OK factorise as 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 attributed to David…

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Definitions

Integral closure
Maximal ideal
Field Field (mathematics)
Dimensional Krull dimension
Chinese remainder theorem
Relative discriminant Discriminant of an algebraic number field
Ideal norm
Prime avoidance lemma
Acts transitively Transitive group action
Orbit Orbit (group theory)
Automorphisms Automorphism
Unique factorisation theorem Fundamental theorem of arithmetic
Orbit-stabilizer formula
Frobenius element
Abelian group
Class field theory
Complex manifolds Complex manifold
Algebraic geometry
Algebraically closed field
Preimages Preimage
Splitting field
Cubic fields Cubic field

Example — the Gaussian integers

Computing the factorisation

Primitive element theorem
Minimal polynomial Minimal polynomial (field theory)
Monic polynomial
Conductor Conductor (ring theory)
Order Order (ring theory)
Conductor Conductor (class field theory)

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Splitting of prime ideals in Galois extensions

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prime field group galois inertia ol integers case ok one extension ideals theory decomposition number primes pj displaystyle given ring

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