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
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
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 and r = dim H². 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; 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. 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.