I am reading this paper and the following quote is confusing me.
For context, we have $x, y \in \Bbb Z[\zeta_p]$ (where $\zeta_p$ is a $p$th root of unity) and $a, b \in \Bbb Z$. Also, we have the following lemma
Let $\alpha \in \Bbb Z[\zeta_p]\setminus (1 - \zeta_p)\Bbb Z[\zeta_p]$. Then there exists $l \in \Bbb Z$ such that $\zeta_p^l\alpha \equiv a \bmod (1 - \zeta_p)^2$ with $a \in \Bbb Z.$
Now this part is confusing me:
We know that $\zeta_p^ix$ and $\zeta_p^jy$ are congruent to rational integers modulo $(1 - \zeta_p)^2$. Since we merely need $x$ and $y$ to satisfy the equation $U(1 - \zeta_p)^{kp}z_0^p = x^p + y^p$, exchanging them for $\zeta_p^i x$ and $\zeta_p^j y$ does not change anything. We know that $x + y \equiv a +b \bmod (1 - \zeta_p)^2$ where $a, b \in \Bbb Z$ are the integers congruent to $x $ and $y$ respectively. Since $1 - \zeta_p \mid x + y$ then $1 - \zeta_p \mid a+ b$, which implies $p \mid a + b$ since $a + b \in \Bbb Z$. This in turn proves that $(1 - \zeta_p)^2 \mid x + y$ which tells us that $k$ must be strictly greater than $1$.
I have a couple of things to ask.
Firstly, the author remarks that $x$ and $y$ can be exchanged for $\zeta_p^ix$ and $\zeta_p^jy$ without changing anything, but then goes on to use $x$ and $y$ in the congruence. Has the author already made this switch and just renamed $\zeta_p^i x$ with $x$, so as to use the lemma?
Secondly, I understand that since $x + y \equiv a + b \bmod (1 - \zeta_p)^2$ and $1 - \zeta_p \mid x + y$, we have that $1-\zeta_p \mid a + b$, and since $(1 - \zeta_p)\Bbb Z[\zeta_p] \cap \Bbb Z = p\Bbb Z$, we have that $p \mid a + b$, but I do not see why we can conclude from this that $(1 - \zeta_p)^2 \mid x + y$ (or at least, I do not see how this can be considered as obvious as the author seems to think it is!)
For the first question: Yes.
For the second: Note that $p = (unit) \cdot (1 - \zeta_p) ^{p-1} $. Since $p \mid a + b$, we know $(1-\zeta_p)^{p-1} \mid a +b$. In particular, $(1-\zeta_p)^2 \mid a +b$ and the congruence implies that $(1-\zeta_p)^2 \mid x+y$ as well.