Hilbert Theorem 90

Let L/K be a finite Galois extensionMathworldPlanetmath with Galois groupMathworldPlanetmath G=Gal(L/K). The modern formulation of Hilbert’s Theorem 90 states that the first Galois cohomology group H1(G,L*) is 0.

The original statement of Hilbert’s Theorem 90 differs somewhat from the modern formulation given above, and is nowadays regarded as a corollary of the above fact. In its original form, Hilbert’s Theorem 90 says that if G is cyclic with generatorPlanetmathPlanetmathPlanetmath σ, then an element xL has norm 1 if and only if


for some yL. Note that elements of the form y/σ(y) are obviously contained within the kernel of the norm map; it is the converse that forms the content of the theorem.

Title Hilbert Theorem 90
Canonical name HilbertTheorem90
Date of creation 2013-03-22 12:10:41
Last modified on 2013-03-22 12:10:41
Owner djao (24)
Last modified by djao (24)
Numerical id 7
Author djao (24)
Entry type Theorem
Classification msc 11R32
Classification msc 11S25
Classification msc 11R34
Synonym Hilbert’s Theorem 90
Synonym Satz 90