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

2 ≤ d1d2 ≤ ...

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

 b_j\ge nb_{j-1} -\sum_{i=2}^{j} b_{j-i} r_i.

As a consequence:

Applications

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; 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,

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

References

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