I am reading Iwasawa's paper "On Galois Groups of Local Fields", because I want to understand more about $U^{(1)}$ as a Galois module. He started his paper with the following data:
$q=p^{f_0}$ is a fixed prime power. $G$ is a finite group of order $n=ef$, generated by elements $\sigma$ and $\tau$, such that $$\sigma^f=1, \, \tau^e=1, \sigma\tau\sigma^{-1}=\tau^q.$$ If such a group exists then we must have $q^f\equiv1\mod{e}$.
Then, for the finite field $\mathbb{F}_{q^f}$, we fix in it a primitive $e$-th root of unity $\eta$. We then make its additive group into a $\mathbb{F}_p[G]$- module by defining $$\sigma a=a^q, \,\tau a=\eta^i a,$$ for any element $a\in\mathbb{F}_{q^f}$. This $G$-module (or $\mathbb{F}_{p}$-representation of $G$) is denoted $A_i$. Everything makes perfect sense to me up to this point. I cannot understand the following few sentenses. He said:
"We denote by $A_i^{\prime}$ the $G$-module over the algebraic closure $\Omega_p$ of $\mathbb{F}_p $ obtained from $A_i$ by extending the scalar field $\mathbb{F}_p $ to $\Omega_p$. "
"$A_i^{\prime}$ then contains an $\Omega_p$-basis $a_{j}^{\prime}$ indexed by rediuse class $j\mod{ff_0}$, such that $\sigma a_{j}^{\prime}=a^{\prime}_{j-f_0}$, $\tau a^{\prime}_j=\eta^{ip^j}a_j^{\prime}$."
My questions are the following:
In the first sentence, did he mean $A_{i}^{\prime}=A_i\otimes_{\mathbb{F_p}}\Omega_p$, with the $G$-action only on $A_i$?
In the second sentense, how did the exponent of $\eta$ change from $i$ to $i\cdot p^j$? I mean I regarded $\eta^i$ as the eigenvalue of the operation $\tau$, which should be invariant under scalar extension and change of basis. So I believe that there must be something I misunderstood.
Any explanation is appreciated. Many thanks!