Whitehead torsion (Ex)

Show that $1 - t - t^{-1} \in \Zz[\Zz/5]$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}1 - t - t^{-1} \in \Zz[\Zz/5]$ for $t\in \Zz/5$$t\in \Zz/5$ the generator is a unit and hence defines an element $\eta$$\eta$ in $\textup{Wh}(\Zz/5)$$\textup{Wh}(\Zz/5)$. Prove that we obtain a well-defined map

$\displaystyle \textup{Wh}(\Zz/5) \to \Rr$ $$\displaystyle \textup{Wh}(\Zz/5) \to \Rr$$

by sending the class represented by the $\Zz[\Zz/5]$$\Zz[\Zz/5]$-automorphism $f \colon \Zz[\Zz/5]^n \to \Zz[\Zz/5]^n$$f \colon \Zz[\Zz/5]^n \to \Zz[\Zz/5]^n$ to $\ln(|\det(\overline{f})|)$$\ln(|\det(\overline{f})|)$, where $\overline{f} \colon \Cc^n \to \Cc^n$$\overline{f} \colon \Cc^n \to \Cc^n$ is the $\Cc$$\Cc$-linear map

$\displaystyle f \otimes_{\Zz[\Zz/5]} \id_{\Cc} \colon \Zz[\Zz/5]^n \otimes_{\Zz[\Zz/5]} \Cc \to \Zz[\Zz/5]^n \otimes_{\Zz[\Zz/5]} \Cc$ $$\displaystyle f \otimes_{\Zz[\Zz/5]} \id_{\Cc} \colon \Zz[\Zz/5]^n \otimes_{\Zz[\Zz/5]} \Cc \to \Zz[\Zz/5]^n \otimes_{\Zz[\Zz/5]} \Cc$$

with respect to the $\Zz/5$$\Zz/5$-action on $\Cc$$\Cc$ given by multiplication with $\exp(2 \pi i/5)$$\exp(2 \pi i/5)$. Finally show that $\eta$$\eta$ generates an infinite cyclic subgroup in $\textup{Wh}(\Zz/5)$$\textup{Wh}(\Zz/5)$.

This is a detailed version of [Milnor1966, Example 6.6] and [Kreck&Lück2005, Ex 5.4].