Introduction to Ideal Class Groups

Algebraic number theory is a really interesting subject, but unlike some other subjects, it’s not 100% clear what objects people study. This post provides an introduction to the class group of a finite dimensional field extension of $\mathbb{Q}$, an object often used in modern number theory.

Fix a finite dimensional field extension $K/\mathbb{Q}$. It turns out that there is a canonical ring associated to $K$, which we’ll denote $O_K$, called the ring of integers of K. Specifically, $O_K$ is defined to be the set of all elements of $K$ which are solutions to monic, integer polynomials. (As a sanity check, one can check the ring of algebraic integers of $\mathbb{Q}$ is $\mathbb{Z}$, which provides motivation for the term for the term “ring of integers”.) For example, the ring of integers of $\mathbb{Q}[\sqrt{-5}]$ is $\mathbb{Z}[\sqrt{-5}]$, but on the other hand, the ring of integers of $\mathbb{Q}[\sqrt{5}]$ is $\mathbb{Z}[\frac{1 + \sqrt{5}}{2}]$.

The next logical step is to ask what properties that rings of algebraic integers have. One might hope that the ring of algebraic integers is a unique factorization domain (UFD). However, in $\mathbb{Z}[\sqrt{-5}],$ we have that $2*3 =  6 = (1 + \sqrt{-5})(1 – \sqrt{-5})$, and it’s not too hard to show that the above equation gives two distinct factorizations of 6.  However, one might notice that when passing to ideals in $\mathbb{Z}[\sqrt{-5}]$, then $(6)$ factors as the product of prime ideals

$(6) = (2, 1 + \sqrt{-5})(2, 1 – \sqrt{-5})(3, 1 + \sqrt{-5})(3, 1 – \sqrt{-5})$

and moreover, this factorization is unique. One can then go onto show that if $O_K$ is the ring of algebraic integers for some finite dimensional field extension $K/\mathbb{Q}$, then for any nonzero ideal $I \subset O_K, I$ can be factored uniquely as the product of prime ideals. This leads to the notion of the Dedekind domain, which generalizes this property.

Moreover, one can argue that one can make a group of these elements by including an extended notion of ideals known as fractional ideals. These are $O_K$-submodules $J \subset K$ for which there is an $r \in O_K$ with $rJ \subset O_K$. This is a group with a product operation similar to that of the rational numbers, so that $\frac{I_1}{J_1}*\frac{I_2}{J_2} := \frac{I_1I_2}{J_1J_2}$.

From this notion, one can define the (ideal) class group of the ring of algebraic integers $O_K$, defined to be the quotient of the above group by the group generated by all nonzero principal ideals. The class group tells us many facts about the associated field and its algebraic integers – it’s a good exercise to check that the ring of algebraic integers is a principal ideal domain if and only if its associated class group is trivial.

One of the first cool facts about this is that the class group is always a finite group! This also develops the subject of class field theory, the study of Galois extensions of $\mathbb{Q}$ whose Galois groups are abelian over $\mathbb{Q}$. This can be used to prove the Kronecker-Weber theorem, which says that for any abelian extension $K/\mathbb{Q}$ (i.e. any Galois extension $K/\mathbb{Q}$ for which $Gal(K/\mathbb{Q})$ is abelian), there is a cyclotomic field containing $K$. In short – the class group of a number field is a rich object worth studying!

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