Leopoldt's Conjecture

In algebraic number theory, Leopoldt's conjecture, introduced by H.-W. Leopoldt (1962, 1975), 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, introduced by H.-W. Leopoldt (1962).

Leopoldt proposed a definition of a p-adic regulator Rp attached to K and a prime number p. The definition of Rp uses an appropriate determinant with entries the p-adic logarithm of a generating set of units of K (up to torsion), in the manner of the usual regulator. The conjecture, which for general K is still open as of 2009, then comes out as the statement that Rp is not zero.

Read more about Leopoldt's Conjecture:  Formulation

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Leopoldt's Conjecture - Formulation
... Leopoldt's conjecture states that the -module rank of the closure of embedded diagonally in is also Leopoldt's conjecture is known in the special case where is an abelian extension of or an abelian extension of an ... Mih─âilescu (2009, 2011) has announced a proof of Leopoldt's conjecture for all CM-extensions of ... As a consequence, Leopoldt's conjecture for those fields is equivalent to their p-adic Dedekind zeta functions having a simple pole at s = 1 ...

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