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.

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

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; Hasse (1926) attributed this question to Furtwangler, though Furtwangler said he had heard it from Schreier.