The Golod–Shafarevich inequality and the class field tower problem
Keywords:
number theory, class field tower problem, pro-p groups, Golod–Shafarevich inequalityAbstract
In this article we present a proof of the class field tower problem. We begin by introducing pro-p groups, explain how to describe them in terms of generators and relations, and present the Golod–Shafarevich inequality, which establishes a criterion for a pro-p group to be infinite. After introducing some notions from algebraic number theory, we apply the Golod–Shafarevich inequality to the class field tower problem. We obtain a criterion for a number field to have an infinite class field tower, and give explicit examples of number fields satisfying this criterion.
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