Trying to understand the definition of Hilbert Class Field

Question

I am trying to understand the definition of the Hilbert Class Field. From here I got this definition.

Given a number field $K$, there exists a unique maximal unramified Abelian extension $L$ of $K$ which contains all other unramified Abelian extensions of $K$. This finite field extension $L$ is called the Hilbert class field of $K$.

I have the following question

1. What exactly is an an unramified extension of $K$ (Give definition)?

Please make the explanation as simple as possible ?

Also, how is it related to the Kronecker-Weber Theorem ?

Answer 1

The field of rational numbers is the fraction field for the ring of integers. Slmilarly every number field is the fraction field for a canonical choice of subring there (the ring of integers of that field).

Unfortunately in that ring analogue of Fundamental Theorem of Arithmetic is false in general. One has a weaker property: uniqueness of every ideal of that rings as a product of prime ideals.

Given two number fields $K,L$ with $K\subset L$ one can ask the following question: given a prime ideal of the ring of integers $K$ take the ideal generated by it in the ring of integers of $L$. This ideal is rarely a prime ideal of $L$. So one writes it as a product of prime ideals of (ring of integers of) $L$. A prime of $K$ is said to be unramified in $L$ if in this factorization there are no repeated prime factors of $L$.

The extension $L$ over $K$ is said to be unramified if every prime of (ring of integers of) $K$ is unramified in $L$. (There is further condition about infinite primes, I am omitting: this is about real embeddings).

For the rational number field every extension is ramified: the prime numbers dividing the discriminant of that field always ramify (discriminant is never $\pm1$).

Class field theory is about existence of unramified extension of a given algebraic number field $K$ that is a Galois extension with abelian Galois group.

Answer 2

Based on your previous participation on this site, I'm going to assume you know what it means for a given prime ideal of $K$ to be ramified in $L$.

An everywhere unramified extension $L/K$ is one which:

• is unramified at every prime ideal of $K$
• is "unramified at the infinite places". Put simply, this means that if we take any real embedding $K\hookrightarrow\mathbb R$ and extend it to an embedding $L\hookrightarrow \mathbb C$, then this embedding has real image.

An example is $\mathbb Q(\sqrt{-5},i)/\mathbb Q(\sqrt{-5})$, where one can check that every prime is unramified (you only need to check the primes dividing the discriminant - i.e. those above $2$ and $5$), and both fields have only complex embeddings, so the extension is unramified at the infinite places by default.

A key point of class field theory is to show that if $H$ is the Hilbert class field of $K$, then $\mathrm{Gal}(H/K)$ is the class group of $K$. This means that $K$ already contains all the information about its abelian unramified extensions. This can be seen as a generalisation of quadratic reciprocity. In particular, there are no non-trivial examples of everywhere unramified abelian extensions of $\mathbb Q$.

Class field theory goes further, and allows you to completely classify the abelian extensions of $K$ which are unramified outside of a finite set of primes. However, this classification is non-explicit: class field theory can say that an extension exists, but does not give a way to construct it. The Kronecker-Weber theorem can be viewed as an example of explicit class field theory over $\mathbb Q$: it allows us to actually construct all the abelian extensions of $\mathbb Q$.