Doubt in a lemma related to Stickeleberger Theorem I am reading the book Introduction to Cyclotomic Fields by Lawrence C. Washington. I have the following doubt:
$\sigma_c$ is an element of Gal$(\mathbb{Q}(\zeta_m)/\mathbb{Q})$ as well as it's restriction to $\mathbb{Q}(\zeta_m)$ and 
$c \in \mathbb{Z}.$ How can we subtract these two elements in the statement of lemma $6.9$?
I thought maybe it's something to do with the group rings but I could not see it how. I studied group rings again from Dummit and Foote's book but could not get anything. Please help.

 A: 
$σ_c$ is an element of $\operatorname{Gal}(\mathbb{Q}(\zeta_m)/\mathbb{Q})$ as well as it's restriction to $\mathbb{Q}(\zeta_m)$

Correction: its restriction to $M$. That restriction is $σ_c|_M : M \to M$ instead of $\mathbb{Q}(\zeta_m) \to \mathbb{Q}(\zeta_m)$. This is an element of $\operatorname{Gal}(M/\mathbb{Q})$.

I thought maybe it's something to do with the group rings but I could not see it how.

That's exactly what it is. So if $G = \{g_0,\dots,g_{n-1}\}$ is a finite group and let's say $g_0$ is the identity, then the elements of $\mathbb{Z}[G]$ look like
$$ \mathbb{Z}[G] = \{ a_0g_0 + a_1g_1 + \dots + a_{n-1}g_{n-1} : a_0,\dots,a_{n-1} \in \mathbb{Z}\}. $$
But it is common to write just "$a_0$" instead of "$a_0g_0$" which comes from identifying $\mathbb{Z}$ with its image in $\mathbb{Z}[G]$. Recall that for any ring $R$ there is exactly one ring map from $\mathbb{Z} \to R$ where $x \mapsto x \cdot 1_R$. In this case the unit of $\mathbb{Z}[G]$ is $g_0$.
So in "$c - \sigma_c$" you are taking $c$ times the identity of $G$ minus $1$ times $\sigma_c$.
