I have a student who has taken some linear algebra but no abstract algebra, and he wants to learn some interesting math.

I explained how, by using companion matrices, one can represent and work with roots of polynomials via matrices. The resulting expanded number system is a finite dimensional vector space over the base field (here just the rationals) and multiplication by elements is actually a linear transformation of that space.

He seemed intrigued, and I would like to know how much more of the wonderful world of number fields he could learn just from the linear algebra perspective, avoiding abstract algebra (e.g. groups and rings). That probably means most Galois theory is out. I think a very light amount of abstract algebra might be okay, but I don't want that to be what he is spending most time learning during this semester's project. Some modular arithmetic would also be fine.

Does anyone know a good reference (book/chapter/article) that might suit this purpose? Or maybe a nice theorem we could study?

Thanks.