Leopoldt's conjecture

For every number field and prime, the closure of its global units inside the product of its local units has rank equal to the number field’s unit rank.

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Number field: A finite extension of the rational numbers. The conjecture ranges over every such field.
Heinrich-Wolfgang Leopoldt: A German mathematician who studied algebraic number theory and formulated Leopoldt's conjecture. His name is attached to the conjecture and its original formulation.
Cyclotomic Zp-extension: The infinite extension obtained by adjoining compatible p-power roots of unity, with Galois group isomorphic to the p-adic integers. Leopoldt's conjecture helps determine the expected rank of its Galois group over a number field.
Leopoldt's conjecture for abelian fields: The assertion that Leopoldt's rank prediction holds for every abelian number field and every prime. This substantial family is established by methods using abelian L-functions.
p-adic number: An element of the completion of the rational numbers for the p-adic absolute value. Local units are formed in completions of the field at primes above p.
p-adic class field theory: The study of abelian extensions of number fields using p-adic methods and local-global reciprocity. Leopoldt's conjecture arose in the setting of p-adic class field theory.
p-adic regulator: A determinant built from p-adic logarithms of fundamental units, whose nonvanishing measures their p-adic independence. Its nonvanishing provides a practical criterion in explicit cases.
Baker–Brumer theorem: A theorem establishing linear independence of logarithms of algebraic numbers over suitable algebraic coefficients in a p-adic setting. Its p-adic linear-independence results prove Leopoldt's conjecture for important classes of fields.
Unit group: The multiplicative group of invertible elements in a ring. Global and local units are the groups whose ranks the conjecture compares.
Local class field theory: A theory relating abelian extensions of a local field to its multiplicative group. Local reciprocity clarifies why local unit groups enter the conjecture.