Let $\Gamma$ be a degree $5$ extension of $\mathbb{Q}$ such that $[\Gamma(\zeta_5): \Gamma] = 4 $ $(\zeta_5\, is\,\, 5^{th}\,\,\, root\,\, of\,\, unity)$ $G= Gal(\Gamma(\zeta_5)/ \Gamma) \cong \mathbb{Z}/4\mathbb{Z}$ such that $Gal(\Gamma/\mathbb{Q}) = \langle\tau\rangle$ and $\tau^4 = 1$. we consider the action of $G$ On a $\mathbb{Z}_5[G]$-module denoted C.
I need to prove that $C \cong C^+\times C^-$ with $C^+ = \{\mathcal{A} \in C | \mathcal{A}^\tau = \mathcal{A}\}$ and $C^- = \{\mathcal{A} \in C | \mathcal{A}^\tau = \mathcal{A}^{-1}\}$.
I had found an element in $C^+$ ( is $\mathcal{A}^{1+\tau+\tau^2+\tau^3}$) but I don't find element in $C^-$, that verify the direct product of $C$