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This paper refers to Langlands' big set of notes [L] devoted to the questionif the (normalized) local Hecke-Tate root number $\Delta=\Delta(E,\chi)$, where$E$ is a finite separable extension of a fixed non-archimedean local field $F$,and $\chi$ a quasicharacter of $E^\times$, can be appropriately extended to alocal $\varepsilon$-factor $\varepsilon_\Delta=\varepsilon_\Delta(E,\rho)$ forall virtual representations $\rho$ of the corresponding Weil group $W_E.$Whereas Deligne [D] has given a relatively short proof by using the globalArtin-Weil L-functions, the proof of Langlands is purely local and splits intotwo parts: the {\bf algebraic part} to find a minimal set of relations for thefunctions $\Delta$, such that the existence (and uniqueness) of$\varepsilon_\Delta$ will follow from these relations; and the more extensive{\bf arithmetic part} to give a direct proof that all these relations areactually fulfilled. Our aim is to cover the algebraic part of Langlands' noteswhich can be done completely in the framework of representations of solvableprofinite groups, where two modifications of Brauer's theorem play a prominentrole.  Introduction / 1. The notion of extendible functions / 2. The kernel of theBrauer map and its generating relations / 3. A criterion for the extendibilityof functions / 4. Recovering Theorem 3.1 for the case of local root numbers /Appendices A1. Proving Brauer 3 and Brauer 4 / A2. On type-III-groups.

Extendible Functions and Local Root Numbers Remarks on a paper of R.P. Langlands