# Fields and Modules in Number Theory?

I'm about to start self-studying number theory and I was wondering if these two topics are essential to number theory on the introductory level, I'm going to study them in college eventually, but I'm wondering if I should learn about them right away if I want to dive into number theory. I studied elementary number theory, and I know group and ring theory. So my questions are:  In what areas of number theory are these two branches used and what parts of them are used? And will I need them in an introductory course in algebraic number theory? Thanks in advance

• There's no way to answer this without knowing your background. There is a lot of material in elementary number theory that doesn't require any heavy algebraic machinery, but eventually you will certainly need the topics you mention. – lulu Jun 9 '17 at 15:15
• ...and you will need them very big time if you study algebraic numbers theory...and pretty much from the beginning. – DonAntonio Jun 9 '17 at 15:17
• Field theory and module theory are essential to algebraic number theory but it varies from course to course how much of this is considered background knowledge or needs to be developed in the course. – Trevor Gunn Jun 9 '17 at 15:22
• See J.P. Lafon's nice motivation for modules in this answer. (hopefully someone will $\TeX$ it) – Bill Dubuque Jun 9 '17 at 20:09
• You can hardly avoid field theory even in elementary number theory. For example: Theorem: Let $(F, +,\times,0,1)$ be a finite field. Then the group $(F$ \ $\{0\},\times,1)$ is cyclic. Corollary: If $p$ is prime there exists integer $x$ such that $x^n\not \equiv 1 \pmod p$ for $1\leq n <p-1.$ – DanielWainfleet Jun 10 '17 at 2:12

## 1 Answer

I taught myself algebraic number theory a few years ago without a really solid understanding of module theory. It was painful. I ragequit quite a few times. I recommend you solidify your algebra background before you start.

I think of basic algebraic number theory as the study of integral closures of $\mathbb{Z}$ (or $\mathbb{Z}_p$) in finite extensions of $\mathbb{Q}$ (or $\mathbb{Q}_p$). That is, you have a field $K$ containing $\mathbb{Q}$, such that $K$ is finite dimensional as a vector space over $\mathbb{Q}$. Then the ring of integers of $K$ is defined to be the set $\mathcal O_K$ of elements of $K$ which are roots of a polynomial in $\mathbb{Z}[X]$ with leading coefficient one. For example, $\mathcal O_{\mathbb{Q}} = \mathbb{Z}$, and $\mathcal O_{\mathbb{Q}(i)} = \{ a + bi : a, b \in \mathbb{Z}\}$.

The first nontrivial results of algebraic number theory are heavily dependent on results from module theory. For example, if $K \subseteq L$ are finite extensions of $\mathbb{Q}$, then $\mathcal O_L$ and $\mathcal O_K$ are free abelian groups, and $\mathcal O_L$ is finitely generated as a module over $\mathcal O_K$. The proof I know of this relies on several nonobvious facts:

1 . If $M$ is a finitely generated module over a Noetherian ring $R$, then any submodule of $M$ is also finitely generated over $R$.

2 . A finitely generated module over a principal ideal domain is free if and only if it is torsion free.

3 . If $V$ is a finite dimensional vector space over a field $k$, and $B$ is a nondegenerate bilinear form on $V$, then to any basis $v_1, ... , v_n$ of $V$ there exists a dual basis $v_1^{\ast}, ... , v_n^{\ast}$ of $V$, such that $B(v_i, v_j^{\ast}) = \delta_{ij}$.

From field theory, you also end up using some not very obvious facts. For example, if $k \subseteq l$ is a finite separable field extension, $y \in l$, and the trace of $xy$ is zero for every $x \in l$, then $y = 0$.

Galois theory is something you need to know really well. You will constantly be looking at subgroups of $\textrm{Gal}(L/K)$ for Galois $L/K$, and relating these subgroups of intermediate fields of $L/K$.

On top of this, you need to have a solid understanding of basic commutative algebra: ideals, quotient rings, localization. Most algebraic number theory books I know of don't expect you to know integral closures right off the bat, but if you're not comfortable with say, localization, I think you will have a really hard time.