# Leopoldt’s conjecture

Let $K$ be a number field, and let $p$ be a rational prime. Then $R_{p}(K)\neq 0$, where $R_{p}(K)$ denotes the $p$-adic regulator (http://planetmath.org/PAdicRegulator) of $K$.

Though unproven for number fields in general, it is known to be true for abelian extensions of $\mathbb{Q}$, and for certain non-abelian 2-extensions of imaginary quadratic extensions of $\mathbb{Q}$.

## References

Title Leopoldt’s conjecture LeopoldtsConjecture 2013-03-22 14:14:28 2013-03-22 14:14:28 mathcam (2727) mathcam (2727) 6 mathcam (2727) Conjecture msc 11R27