Leopoldt's conjecture, in algebraic number theory, was introduced by Heinrich-Wolfgang Leopoldt and states that the p-adic regulator of a number field does not vanish. The p-adic regulator is an analogue of the usual regulator, defined using p-adic logarithms instead of the usual logarithms. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/
Facts
StatementStates that the p-adic regulator of a number field does not vanish; the p-adic regulator is defined analogously to the ordinary regulator but using p-adic logarithms in place of ordinary logarithms. 1 Progress Toward ResolutionKnown to hold when the number field is an abelian extension of the rationals or of an imaginary quadratic field, via a reduction by Ax to a p-adic form of a theorem of Baker, later proved by Brumer; Mihailescu has announced a proof for all CM-extensions of the rationals. No general proof exists. 1 Partially Attested
Proposed YearThe year is taken from the 1962 paper by Leopoldt that the article cites where the conjecture is introduced; the article also cites a 1975 paper by Leopoldt there and has no sentence giving the year in words. Classification
Resolution Status Prize Status
Prize Status (category) Connections
In Branch
Source Leopoldt's Conjecture (Wikipedia)
Sources
1. Leopoldt's Conjecture (Wikipedia)
Wikimedia FoundationLead section
states that the p-adic regulator of a number field does not vanish
Lead section, first sentence
Leopoldt's conjecture, introduced by Heinrich-Wolfgang Leopoldt, states that the p-adic regulator of a number field does not vanish.
Formulation section, final paragraph
Leopoldt's conjecture is known in the special case where K is an abelian extension of Q or an abelian extension of an imaginary quadratic number field: Ax reduced the abelian case to a p-adic version of Baker's theorem, which was proved shortly afterwards by Brumer.
References section, Leopoldt 1962 entry
Leopoldt, Heinrich-Wolfgang (1962)
In Branch: Algebraic Number Theory, Lead sentence
In algebraic number theory, Leopoldt's conjecture, introduced by Heinrich-Wolfgang Leopoldt, states that the p-adic regulator of a
View the Source Reader Challenges (0)
No disputes yet. Spotted an error or a better source? Open the first one.
Sign in to dispute this or suggest a correction.