Class field theory facts for kids

Class field theory (CFT) is a big idea in mathematics, especially in a part called algebraic number theory. It helps us understand special ways to "grow" number systems. Imagine you have a basic set of numbers, like all the whole numbers. Class field theory helps explain how to create bigger, more complex number systems from these, especially those where the "growth" follows simple, predictable rules. It uses special tools related to the original number system to describe these new, larger ones.

Many mathematicians helped develop these ideas over decades. David Hilbert is often seen as a pioneer, but Leopold Kronecker and Eduard Ritter von Weber also played key roles. Weber even coined the term "class field." Later, Teiji Takagi and Emil Artin proved many important ideas in this theory.

One of the main results is about a number system F. If you create its "maximal abelian unramified extension" (a special, simple way to grow it, called K), then the way K relates to F (its Galois group) is directly connected to something called the ideal class group of F. This idea was made even bigger with the Artin reciprocity law. This law connects the "growth" of a number system to another special group called the idele class group.

The "existence theorem" is another key part. It says that the tools from the Artin reciprocity law can be used to match up all possible simple "growths" of a number system with certain subgroups of its idele class group.

Since the 1930s, mathematicians often built this theory by first understanding "local class field theory" (which looks at numbers very closely) and then using it to build "global class field theory" (which looks at numbers more broadly). Emil Artin and John Tate did this using a method called group cohomology. Later, Jürgen Neukirch found a way to prove the main ideas without using cohomology, making it more direct.

What is Class Field Theory?

In modern math, class field theory describes the "maximal" simple extension of a number system. This extension is infinitely large. The goal of class field theory is to describe this huge extension using simpler, more manageable objects from the original number system. It also aims to create a direct link between finite (not infinite) simple extensions and certain subgroups of these objects. This link is so strong that for every such subgroup, there's a unique "class field" (a specific extension), which is where the theory gets its name.

For certain simple number systems, like the rational numbers (our everyday fractions), there's a very detailed theory. For example, the biggest simple extension of rational numbers is created by adding all possible "roots of unity" (numbers that become 1 when raised to a certain power). This is known as the Kronecker–Weber theorem. This theorem helps explain the Artin reciprocity map in a very clear way for rational numbers. However, these detailed methods don't work for all number systems, so mathematicians use different ideas for the general theory.

One common way to build the main connections in class field theory is to first understand the "local" connections (how numbers behave very closely). Then, mathematicians prove that when you combine all these local connections for a "global" number system, they behave in a special way. This special behavior is called the "global reciprocity law." It's a much broader version of an old idea called the quadratic reciprocity law discovered by Gauss.

Important Discoveries

The ideas behind class field theory started with the quadratic reciprocity law proved by Gauss. Over time, many mathematicians contributed, including Ernst Kummer and Leopold Kronecker, who worked on ideas related to numbers and their "completions."

The first very clear class field theories were for specific cases, like using "roots of unity" for rational numbers or "elliptic curves" for certain complex numbers. Later, Goro Shimura provided another clear theory for a different group of number systems.

However, these very clear theories couldn't be used for all number systems. So, the general class field theory had to use different ideas that work for every "global field" (a broad type of number system).

The famous problems posed by David Hilbert encouraged more development. This led to the reciprocity laws and proofs by Teiji Takagi, Phillip Furtwängler, Emil Artin, Helmut Hasse, and many others. By 1920, Takagi's important "existence theorem" was known, and most of the main results were established by 1930.

In the 1930s, mathematicians started using "infinite extensions" and Wolfgang Krull's ideas about their Galois groups. This, combined with Pontryagin duality, made the main result (the Artin reciprocity law) clearer, though more abstract. A big step was when Claude Chevalley introduced "ideles" in the 1930s. Ideles helped simplify how abelian extensions of global fields are described. By 1940, most of the main results were proven.

Later, the results were rephrased using group cohomology, which became a common way to learn class field theory. However, this method was not always very direct. In the 1990s, thanks to work by Bernard Dwork, John Tate, Michiel Hazewinkel, and Jürgen Neukirch, a very clear way to understand class field theory without using cohomology was developed.

Why is it Important?

Class field theory is used in many areas of algebraic number theory. For example, it helps prove something called Artin-Verdier duality. The very clear versions of class field theory are used in areas like Iwasawa theory and Galois modules theory.

Many big achievements in math, like parts of the Langlands correspondence and BSD conjecture, use methods from class field theory.

Generalizations of Class Field Theory

There are three main ways mathematicians have tried to expand class field theory:

• The Langlands program
• Anabelian geometry
• Higher class field theory

The Langlands program is sometimes seen as a "non-abelian" version of class field theory. It aims to describe more complex types of number system extensions. However, it doesn't include all the detailed information that class field theory does for the simpler "abelian" cases. It also doesn't have an "existence theorem" like class field theory, meaning the idea of "class fields" isn't present in the Langlands program.

Anabelian geometry is another expansion. It studies how to figure out the original number system or other mathematical objects just by knowing their "Galois group" or "fundamental group."

Higher class field theory is a natural expansion that describes simple extensions of "higher local fields" and "higher global fields." These are even more complex types of number systems. This theory uses advanced tools from algebraic K-theory.

See also

• Non-abelian class field theory
• Anabelian geometry
• Frobenioid
• Langlands correspondences