TITLE: Special topics to include in course in algebraic number theory
QUESTION [17 upvotes]: I'll be teaching an introductory course in algebraic number theory this fall (stopping before class field theory).  I'm looking for a good list of "special topics" I can include to illustrate the general theory.  In other words, attractive theorems (preferably off the beaten path) that can be proved using the basic results and that illustrate their power.  For example, one standard one is the proof of quadratic reciprocity using cyclotomic fields.  Can people suggest other ones?  Especially ones that connect to other branches of mathematics (e.g. combinatorics, geometry/topology, group theory, etc)?

REPLY [7 votes]: Here's something within algebraic number theory, but is sort of a "special topic" in the sense that it's not treated in most algebraic number theory courses, but could easily be.  It also has some combinatorial aspects. Everyone knows that the class group of a number field $K$ measures the failure of unique factorization into irreducibles in its ring of integers.  
Question: what does this really mean?  That is, in what way does it quantitatively measure this?  
A sample result is Carlitz's theorem, which says all irreducible factorizations of a given element have the same length if and only if the class number is at most 2.  You can actually count the number of factorizations of $x$ based on the prime ideal factorization of $(x)$ in $\mathcal O_K$.  I discussed this in my course a few years ago, based on this note.  The note make more clear connections with combinatorics.  Narkiewicz's book also discusses this problem from a different perspective.