In abstract algebra, Hilbert's Theorem 90 (or Satz 90) is an important result on cyclic extensions of fields (or to one of its generalizations) that leads to Kummer theory.
In its most basic form, it states that if L/K is an extension of fields with cyclic Galois group G = Gal(L/K) generated by an element
is an element of L of relative norm 1, that is
The theorem takes its name from the fact that it is the 90th theorem in David Hilbert's Zahlbericht (Hilbert 1897, 1998), although it is originally due to Kummer (1855, p.213, 1861).
Often a more general theorem due to Emmy Noether (1933) is given the name, stating that if L/K is a finite Galois extension of fields with arbitrary Galois group G = Gal(L/K), then the first cohomology group of G, with coefficients in the multiplicative group of L, is trivial: Let
The Galois group is cyclic of order 2, its generator
acting via conjugation: An element
An element of norm one thus corresponds to a rational solution of the equation
or in other words, a point with rational coordinates on the unit circle.
Hilbert's Theorem 90 then states that every such element a of norm one can be written as where
is as in the conclusion of the theorem, and c and d are both integers.
This may be viewed as a rational parametrization of the rational points on the unit circle.
The theorem can be stated in terms of group cohomology: if L× is the multiplicative group of any (not necessarily finite) Galois extension L of a field K with corresponding Galois group G, then Specifically, group cohomology is the cohomology of the complex whose i-cochains are arbitrary functions from i-tuples of group elements to the multiplicative coefficient group,
( ϕ ) ) ( σ , τ )
ϕ ( σ τ ) ,
under the action of the group element
, with its unique image value
The triviality of the first cohomology group is then equivalent to the 1-cocycles
ϕ ( σ τ ) = ϕ ( σ )
{\displaystyle {\begin{array}{rcl}Z^{1}&=&\ker d^{1}&=&\{\phi \in C^{1}{\text{ satisfying }}\,\,\forall \sigma ,\tau \in G\,\colon \,\,\phi (\sigma \tau )=\phi (\sigma )\,\phi (\tau )^{\sigma }\}\\{\text{ is equal to }}\\B^{1}&=&{\text{im }}d^{0}&=&\{\phi \in C^{1}\ \,\colon \,\,\exists \,b\in L^{\times }{\text{ such that }}\phi (\sigma )=b/b^{\sigma }\ \ \forall \sigma \in G\}.\end{array}}}
On the other hand, a 1-coboundary is determined by
Equating these gives the original version of the Theorem.
A further generalization is to cohomology with non-abelian coefficients: that if H is either the general or special linear group over L, including
is the group of isomorphism classes of locally free sheaves of
-modules of rank 1 for the Zariski topology, and
is the sheaf defined by the affine line without the origin considered as a group under multiplication.
[1] There is yet another generalization to Milnor K-theory which plays a role in Voevodsky's proof of the Milnor conjecture.
of norm By clearing denominators, solving
-vector spaces via The primitive element theorem gives
has minimal polynomial we can identify via Here we wrote the second factor as a