# Number fields without abstract algebra

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.

• Take a look at mathoverflow.net/questions/275767/… and links there. The part that I like is that we get a homogeneous polynomial with integer coefficients that is completely multiplicative, Sep 12 '17 at 20:56
• Kummer did his work on cyclotomic fields before the abstract notions of rings and fields were introduced. For an exposition of Kummer's work more or less in classical language see Edwards "Fermat's Last Theorem". Sep 12 '17 at 21:18
• I figured out the Prime Ideal Sep 12 '17 at 23:18
• IMO abstract algebra is truly needed only in the proof of the unique factorization of ideals, in the surjection $G$ to the decomposition group, and when listing the transitive subgroups of $S_n$ (I know it because I tried to avoid AA as possible) Sep 12 '17 at 23:26