# Golod–Shafarevich theorem

In mathematics, the **Golod–Shafarevich theorem** was proved in 1964 by two Russian mathematicians, Evgeny Golod and Igor Shafarevich. It is a result in non-commutative homological algebra which has consequences in various branches of algebra.

## The inequality

Let *A* = *K*<*x*_{1}, ..., *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 ≤
*d*_{1}≤*d*_{2}≤ ...

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}≤*d*^{2}/4 for all*i*

- if
*B*is finite-dimensional, then*r*_{i}>*d*^{2}/4 for some*i*.

## Applications

This result has important applications in combinatorial group theory:

- If
*G*is a nontrivial finite p-group, then*r*>*d*^{2}/4 where*d*= dim*H*^{1}(*G*,**Z**/*p***Z**) and*r*= dim*H*^{2}(*G*,**Z**/*p***Z**) (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*>*d*^{2}/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. Another consequence of the construction 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

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- Johnson, D.L. (1980). "Topics in the Theory of Group Presentations" (1st ed.). Cambridge University Press. ISBN 0-521-23108-6. See chapter VI.
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- 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.)