I am trying to understand the construction done in this reference http://www.personal.psu.edu/sxk37/pub/KKS-old.pdf by Katok, Katok and Schmidt.
In the section 3.3 (p11-13), the idea is to characterize all irreducible actions of $\mathbb{Z}^d$ by automorphisms on the torus $\mathbb{T}^n = \mathbb{R}^n/\mathbb{Z}^n$ up to algebraic conjugacy.
Given such an action $\alpha$ generated by matrices $A_1,\cdots, A_d$, one can construct a field extension $\mathbb{Q}(\lambda)$ of degree $n$, together with a $d$-tuple $\overline{\lambda} = (\lambda_1,\cdots,\lambda_d)$ of multiplicatively independant units in $\mathbb{Q}(\lambda)$, and a lattice $\mathcal{L}$ which is a $\mathbb{Z}[\lambda_1,\cdots,\lambda_d]$-module, such that the original action $\alpha$ is algebraically conjugate to the action of $\mathbb{Z}^d$ on $\mathbb{Q}(\lambda)$ given by $(k_1,\cdots,k_d).x = (\Pi_{i=1}^d \lambda_i^{k_i})x$. I understand how to construct all these but I have a problem concerning the converse construction.
Starting from a field extension $K$ together with the tuple of units $\overline{\lambda}$, and the lattice $\mathcal{L}$ which is also a $\mathbb{Z}[\lambda_1,\cdots,\lambda_d]$-module, we want to construct the corresponding action $\alpha_{\overline{\lambda},\mathcal{L}}$, which is the inverse of the previous construction. This is clear for me how to do, by posing $A_i = (\pi^{-1}(\lambda_iv_1),\cdots,\pi^{-1}(\lambda_iv_n))$, where $v_i$ is a basis of $\mathcal{L}$, and $\pi : \mathbb{Q}\to K$ is given by $(r_1\cdots r_n)\mapsto \sum_{i=1}^n r_iv_i$. I checked with the examples at the end of the article, and it matches their results, so I am assuming that my definition of $A_i$ is correct. However I do not understand if the action $\alpha_{\overline{\lambda},\mathcal{L}}$ obtained is irreducible in general. The article is vague about this point "We will assume that the action is irreducible which in many interesting cases can be easily checked", but does not refer to any irreducibility checking later on. Is it true that the actions obtained like this are always irreducible, or is it something to be checked in every specific case?