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The so-called Lifted Root Number Conjecture is a strengthening of Chinburg’s $\Omega \left(3\right)$- conjecture for Galois extensions $K/F$ of number fields. It is certainly more difficult than the $\Omega \left(3\right)$-localization. Following the lead of Ritter and Weiss, we prove the Lifted Root Number Conjecture for the case that $F=\mathbb{Q}$ and the degree of $K/F$ is an odd prime, with another small restriction on ramification. The very explicit calculations with cyclotomic units use trees and some classical combinatorics for bookkeeping. An important...