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,..., xn⟩ be the free algebra over a field K in n = d + 1 non-commuting variables xi.
Let J be the 2-sided ideal of A generated by homogeneous elements fj of A of degree dj with
where dj tends to infinity. Let ri be the number of dj equal to i.
Let B=A/J, a graded algebra. Let bj = dim Bj.
The fundamental inequality of Golod and Shafarevich states that
As a consequence:
This result has important applications in combinatorial group theory:
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,
More generally, a number field with sufficiently many prime factors in the discriminant has an infinite class field tower.