I have been studying algebraic number theory from "$p$-adic Numbers : An Introduction" by Gouvea, along with that some other articles also.
One question has come to my mind, I am trying to express it neatly in step-by-step.
Let $\mathbb Q_7$ to be the field of $7$-adic rationals and $\mathbb Z_7$ to be its ring of integers.
It is also know that all primitive $6$-th roots of unity is in $\mathbb Q_7$ [See Section 4.3 of the above mentioned book]. Let us take $\omega$ to be one primitive $6$-th root of unity.
Let $\theta$ be a primitive $7$-th root of unity. Consider the group ($\mathfrak p^2, +)$ where $\mathfrak p^2 = (\theta-1)^2 \mathbb Z_7[\theta]$ is an ideal of $\mathbb Z_7[\theta]$.
Also suppose $\sigma_3$ denotes the ring automorphism of $\mathbb Z_7[\theta]$ defined by $\theta\mapsto \theta^3$ which is $\mathbb Q_7$-linear and it has order $6$. Hence it is diagonisable with eigen values $\{\omega^i\mid 0\le i\le 5\}$.
This allows us to define a $\mathbb Q_7$-linear map $\tau : \mathfrak p^2 \to \mathfrak p^2$ by $x\mapsto \sigma_3(x)+\sigma_3^5(x)-x$ having the eigen values $\{\omega^i+\omega^{5i}-1\mid 0\le i\le 5\} =\{1,-2,-3,0\}$ with the algebraic multiplicities $1,2,1,2$ resctively.
Now my question is
Is there any method so that we can describe the image of $\tau$, more specifically can we find the generators of the abelian group ($\mathfrak p^2/\tau(\mathfrak p^2)$, +) ?
Sorry for not showing much effort from my side. I kind of understand the existence of eigen value zero might cause problem but I am not sure. Any help will be greatly appreciated.
Thanks in advance.