Hilbert's Theorem 90

Hilbert's Theorem 90   (Redirected from Hilbert's theorem 90) Jump to navigation Jump to search 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 {displaystyle sigma ,} and if {displaystyle a} is an element of L of relative norm 1, that is {displaystyle N(a):=a,sigma (a),sigma ^{2}(a)cdots sigma ^{n-1}(a)=1,} then there exists {displaystyle b} in L such that {displaystyle a=b/sigma (b).} 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: {displaystyle H^{1}(G,L^{times })={1}.} Contents 1 Examples 2 Cohomology 3 Proof 4 References 5 External links Examples Let {displaystyle L/K} be the quadratic extension {displaystyle mathbb {Q} (i)/mathbb {Q} } . The Galois group is cyclic of order 2, its generator {displaystyle sigma } acting via conjugation: {displaystyle sigma :c+dimapsto c-di.} An element {displaystyle a=x+yi} in {displaystyle mathbb {Q} (i)} has norm {displaystyle asigma (a)=x^{2}+y^{2}} . An element of norm one thus corresponds to a rational solution of the equation {displaystyle x^{2}+y^{2}=1} 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 {displaystyle a={frac {c-di}{c+di}}={frac {c^{2}-d^{2}}{c^{2}+d^{2}}}-{frac {2cd}{c^{2}+d^{2}}}i,} where {displaystyle b=c+di} 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. Rational points {displaystyle (x,y)=(p/r,q/r)} on the unit circle {displaystyle x^{2}+y^{2}=1} correspond to Pythagorean triples, i.e. triples {displaystyle (p,q,r)} of integers satisfying {displaystyle p^{2}+q^{2}=r^{2}} .

Cohomology 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 {displaystyle H^{1}(G,L^{times })={1}.} 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, {displaystyle C^{i}(G,L^{times })={phi :G^{i}to L^{times }}} , with differentials {displaystyle d^{i}:C^{i}to C^{i+1}} defined in dimensions {displaystyle i=0,1} by: {displaystyle (d^{0}(b))(sigma )=b/b^{sigma },quad {text{ and }}quad (d^{1}(phi ))(sigma ,tau ),=,phi (sigma )phi (tau )^{sigma }/phi (sigma tau ),} where {displaystyle x^{g}} denotes the image of the {displaystyle G} -module element {displaystyle x} under the action of the group element {displaystyle gin G} . Note that in the first of these we have identified a 0-cochain {displaystyle gamma =gamma _{b}:G^{0}=id_{G}to L^{times }} , with its unique image value {displaystyle bin L^{times }} . The triviality of the first cohomology group is then equivalent to the 1-cocycles {displaystyle Z^{1}} being equal to the 1-coboundaries {displaystyle B^{1}} , viz.: {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 ,bin L^{times }{text{ such that }}phi (sigma )=b/b^{sigma } forall sigma in G}.end{array}}} For cyclic {displaystyle G={1,sigma ,ldots ,sigma ^{n-1}}} , a 1-cocycle is determined by {displaystyle phi (sigma )=ain L^{times }} , with {displaystyle phi (sigma ^{i})=a,sigma (a)cdots sigma ^{i-1}(a)} and: {displaystyle 1=phi (1)=phi (sigma ^{n})=a,sigma (a)cdots sigma ^{n-1}(a)=N(a).} On the other hand, a 1-coboundary is determined by {displaystyle phi (sigma )=b/b^{sigma }} . 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 {displaystyle operatorname {GL} _{1}(L)=L^{times }} , then {displaystyle H^{1}(G,H)={1}.} Another generalization is to a scheme X: {displaystyle H_{text{et}}^{1}(X,mathbb {G} _{m})=H^{1}(X,{mathcal {O}}_{X}^{times })=operatorname {Pic} (X),} where {displaystyle operatorname {Pic} (X)} is the group of isomorphism classes of locally free sheaves of {displaystyle {mathcal {O}}_{X}^{times }} -modules of rank 1 for the Zariski topology, and {displaystyle mathbb {G} _{m}} 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.

Proof Let {displaystyle L/K} be cyclic of degree {displaystyle n,} and {displaystyle sigma } generate {displaystyle operatorname {Gal} (L/K)} . Pick any {displaystyle ain L} of norm {displaystyle N(a):=asigma (a)sigma ^{2}(a)cdots sigma ^{n-1}(a)=1.} By clearing denominators, solving {displaystyle a=x/sigma ^{-1}(x)in L} is the same as showing that {displaystyle asigma ^{-1}(cdot ):Lto L} has {displaystyle 1} as an eigenvalue. We extend this to a map of {displaystyle L} -vector spaces via {displaystyle {begin{cases}1_{L}otimes asigma ^{-1}(cdot ):Lotimes _{K}Lto Lotimes _{K}L\ell otimes ell 'mapsto ell otimes asigma ^{-1}(ell ').end{cases}}} The primitive element theorem gives {displaystyle L=K(alpha )} for some {displaystyle alpha } . Since {displaystyle alpha } has minimal polynomial {displaystyle f(t)=(t-alpha )(t-sigma (alpha ))cdots left(t-sigma ^{n-1}(alpha )right)in K[t],} we can identify {displaystyle Lotimes _{K}L{stackrel {sim }{to }}Lotimes _{K}K[t]/f(t){stackrel {sim }{to }}L[t]/f(t){stackrel {sim }{to }}L^{n}} via {displaystyle ell otimes p(alpha )mapsto ell left(p(alpha ),p(sigma alpha ),ldots ,p(sigma ^{n-1}alpha )right).} Here we wrote the second factor as a {displaystyle K} -polynomial in {displaystyle alpha } .

Under this identification, our map becomes {displaystyle {begin{cases}asigma ^{-1}(cdot ):L^{n}to L^{n}\ell left(p(alpha ),ldots ,p(sigma ^{n-1}alpha ))mapsto ell (ap(sigma ^{n-1}alpha ),sigma ap(alpha ),ldots ,sigma ^{n-1}ap(sigma ^{n-2}alpha )right).end{cases}}} That is to say under this map {displaystyle (ell _{1},ldots ,ell _{n})mapsto (aell _{n},sigma aell _{1},ldots ,sigma ^{n-1}aell _{n-1}).} {displaystyle (1,sigma a,sigma asigma ^{2}a,ldots ,sigma acdots sigma ^{n-1}a)} is an eigenvector with eigenvalue {displaystyle 1} iff {displaystyle a} has norm {displaystyle 1} .

References ^ Milne, James S. (2013). "Lectures on Etale Cohomology (v2.21)" (PDF). p. 80. Hilbert, David (1897), "Die Theorie der algebraischen Zahlkörper", Jahresbericht der Deutschen Mathematiker-Vereinigung (in German), 4: 175–546, ISSN 0012-0456 Hilbert, David (1998), The theory of algebraic number fields, Berlin, New York: Springer-Verlag, ISBN 978-3-540-62779-1, MR 1646901 Kummer, Ernst Eduard (1855), "Über eine besondere Art, aus complexen Einheiten gebildeter Ausdrücke.", Journal für die reine und angewandte Mathematik (in German), 50: 212–232, doi:10.1515/crll.1855.50.212, ISSN 0075-4102 Kummer, Ernst Eduard (1861), "Zwei neue Beweise der allgemeinen Reciprocitätsgesetze unter den Resten und Nichtresten der Potenzen, deren Grad eine Primzahl ist", Abdruck aus den Abhandlungen der Kgl. Akademie der Wissenschaften zu Berlin (in German), Reprinted in volume 1 of his collected works, pages 699–839 Chapter II of J.S. Milne, Class Field Theory, available at his website [1]. Neukirch, Jürgen; Schmidt, Alexander; Wingberg, Kay (2000), Cohomology of Number Fields, Grundlehren der Mathematischen Wissenschaften, vol. 323, Berlin: Springer-Verlag, ISBN 978-3-540-66671-4, MR 1737196, Zbl 0948.11001 Noether, Emmy (1933), "Der Hauptgeschlechtssatz für relativ-galoissche Zahlkörper.", Mathematische Annalen (in German), 108 (1): 411–419, doi:10.1007/BF01452845, ISSN 0025-5831, Zbl 0007.29501 Snaith, Victor P. (1994), Galois module structure, Fields Institute monographs, Providence, RI: American Mathematical Society, ISBN 0-8218-0264-X, Zbl 0830.11042 External links Wikisource has original text related to this article: Hilbert's Theorem 90 in: David Hilbert, Gesammelte Abhandlungen, Erster Band Categories: Theorems in algebraic number theory