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:
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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
 A: 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.
