Special topics to include in course in algebraic number theory

Question

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)?

Answer 1

Here are a few suggestions, though within number theory.

• Chebotarev's original proof of the density theorem (in the logarithmic form). Or at least the infinitude of totally split primes as well as the infinitude of non-split primes in every extension $L/K$ of number fields.

• The Erdos support theorem on the multiplicative group: If $p \mid a^n - 1 \Rightarrow p \mid b^n-1$, then $b = a^j$. Local-global statements such as "$a$ is a cube mod all $p \, \Longrightarrow \,$ $a$ is, indeed, a cube."

• Shafarevich's proof of the local Kronecker-Weber theorem, as given in Narkiewicz's book. The global theorem then follows immediately, by Minkowski.

• The use of Frobenius elements and ramification (or of Kronecker-Weber) to prove that $\mathbb{Q}^{\mathrm{ab}}$ has the Bogomolov property: the absolute logarithmic height is bounded below by a positive constant, apart from the roots of unity. More interesting than the result itself result is its corollary resulting in a beautifully transparent proof of Smyth's theorem: $m(P) = \int_{S^1} \log{|P(\theta)|} \, d\theta \geq \mathrm{constant} > 0$ for non-reciprocal $P \in \mathbb{Z}[X]$ other than $0, \pm 1$ or $X-1$. This is due to Amoroso and Dvornicich (A lower bound on the height in abelian extensions), and it is in the fourth chapter of Bombieri and Gubler's book. Amoroso's work gives a few further consequences related to class groups.

• Galois groups of irreducible trinomials. Frequently those are the full symmetric group. $X^n - X - 1$ as an example of an explicit polynomial with group $S_n$. Given the irreducibility of that polynomial (Selmer), this is a good application of Minkowski's theorem: Since $\mathbb{Q}$ has no unramified extensions, the group of every Galois extension $K/\mathbb{Q}$ is generated by its inertia subgroups. But $\alpha^n - \alpha - 1 = n\alpha^{n-1} - 1 = 0$ implies $\alpha = n/(1-n)$, hence our polynomial has most one double root at each prime: all the non-trivial inertia groups are generated by a transposition in $S_n$. This and the transitivity (irreducibility) easily show the group is $S_n$. A reference is Serre's course in Galois theory, whose section 4.4 contains other good examples of Galois groups.

Answer 2

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.