Golod–Shafarevich theorem

In mathematics, the Golod–Shafarevich theorem was proved in 1964 by Evgeny Golod and Igor Shafarevich. It is a result in non-commutative homological algebra which solves the class field tower problem, by showing that class field towers can be infinite.

The inequality

Let A = K⟨x₁, ..., x_n⟩ be the free algebra over a field K in n = d + 1 non-commuting variables x_i.

Let J be the 2-sided ideal of A generated by homogeneous elements f_j of A of degree d_j with

2 ≤ d₁ ≤ d₂ ≤ ...

where d_j tends to infinity. Let r_i be the number of d_j equal to i.

Let B=A/J, a graded algebra. Let b_j = dim B_j.

The fundamental inequality of Golod and Shafarevich states that

b_{j}\geq nb_{j-1}-\sum _{i=2}^{j}b_{j-i}r_{i}.

As a consequence:

B is infinite-dimensional if r_i ≤ d²/4 for all i

Applications

This result has important applications in combinatorial group theory:

If G is a nontrivial finite p-group, then r > d²/4 where d = dim H¹(G,Z/pZ) and r = dim H²(G,Z/pZ) (the mod p cohomology groups of G). In particular if G is a finite p-group with minimal number of generators d and has r relators in a given presentation, then r > d²/4.
For each prime p, there is an infinite group G generated by three elements in which each element has order a power of p. The group G provides a counterexample to the generalised Burnside conjecture: it is a finitely generated infinite torsion group, although there is no uniform bound on the order of its elements.

In class field theory, the class field tower of a number field K is created by iterating the Hilbert class field construction. The class field tower problem asks whether this tower is always finite; Hasse (1926) attributed this question to Furtwangler, though Furtwangler said he had heard it from Schreier. Another consequence of the Golod–Shafarevich theorem is that such towers may be infinite (in other words, do not always terminate in a field equal to its Hilbert class field). Specifically,

Let K be an imaginary quadratic field whose discriminant has at least 6 prime factors. Then the maximal unramified 2-extension of K has infinite degree.

More generally, a number field with sufficiently many prime factors in the discriminant has an infinite class field tower.

References

Golod, E.S; Shafarevich, I.R. (1964), "On the class field tower", Izv. Akad. Nauk SSSR, 28: 261–272 (in Russian) MR0161852
Hasse, Helmut (1926), "Bericht über neuere Unterschungen und Probleme aus der Theorie der algebraischen Zahlkörper.", Jahresbericht der Deutschen Mathematiker-Vereinigung, 35, Göttingen: Teubner
Golod, E.S (1964), "On nil-algebras and finitely approximable p-groups.", Izv. Akad. Nauk SSSR, 28: 273–276 (in Russian) MR0161878
Herstein, I.N. (1968). Noncommutative rings. Carus Mathematical Monographs. MAA. ISBN 0-88385-039-7. See Chapter 8.
Johnson, D.L. (1980). "Topics in the Theory of Group Presentations" (1st ed.). Cambridge University Press. ISBN 0-521-23108-6. See chapter VI.
Koch, Helmut (1997). Algebraic Number Theory. Encycl. Math. Sci. Vol. 62 (2nd printing of 1st ed.). Springer-Verlag. p. 180. ISBN 3-540-63003-1. Zbl 0819.11044.
Narkiewicz, Władysław (2004). Elementary and analytic theory of algebraic numbers. Springer Monographs in Mathematics (3rd ed.). Berlin: Springer-Verlag. p. 194. ISBN 3-540-21902-1. Zbl 1159.11039.
Roquette, Peter (1986) [1967]. "On class field towers". In Cassels, J. W. S.; Fröhlich, A. (eds.). Algebraic number theory, Proceedings of the instructional conference held at the University of Sussex, Brighton, September 1–17, 1965 (Reprint of the 1967 original ed.). London: Academic Press. pp. 231–249. ISBN 0-12-163251-2.
Serre, J.-P. (2002), "Galois Cohomology," Springer-Verlag. ISBN 3-540-42192-0. See Appendix 2. (Translation of Cohomologie Galoisienne, Lecture Notes in Mathematics 5, 1973.)