Algebraic number theory facts for kids

The front page of Disquisitiones Arithmeticae, a very important book by Carl Friedrich Gauss.

Algebraic number theory is a part of number theory that uses tools from abstract algebra to study numbers like integers (whole numbers), rational numbers (fractions), and their more complex relatives. It helps us understand special kinds of numbers called algebraic number fields and their rings of integers.

This field of math looks at properties of these numbers, such as whether they can be broken down into unique prime factors, how certain groups of numbers called ideals behave, and how Galois groups work. These studies can help solve big questions in number theory, like finding solutions to Diophantine equations (equations where you're only looking for whole number answers).

History of Algebraic Number Theory

Diophantus and Ancient Problems

The journey of algebraic number theory began with Diophantus, a mathematician from ancient Alexandria (around 3rd century AD). He studied equations that are now named after him: Diophantine equations. These are equations where you only want whole number solutions.

For example, a famous Diophantine problem is finding whole numbers x and y where:

A = x + y
B = x² + y²

People have studied these types of problems for thousands of years. The Pythagorean triples (like 3, 4, 5, where 3² + 4² = 5²) are solutions to x² + y² = z². The Babylonians (around 1800 BC) already knew about these! Also, the Euclidean algorithm (from about 5th century BC) can help solve simple Diophantine equations like 26x + 65y = 13.

Diophantus's main book was called Arithmetica.

Fermat's Challenge

Pierre de Fermat was a French mathematician who, in 1637, wrote a famous note in the margin of his copy of Arithmetica. He claimed he had a proof for what is now known as Fermat's Last Theorem: that there are no whole numbers a, b, c that can solve aⁿ + bⁿ = cⁿ if n is a whole number greater than 2. He famously said the margin was too small for his proof.

For 358 years, mathematicians tried to prove it, but no one could! This unsolved problem pushed the development of algebraic number theory in the 1800s and led to the proof of the modularity theorem in the 1900s. Finally, in 1995, Andrew Wiles published a successful proof.

Gauss Organizes Number Theory

One of the most important books in algebraic number theory is Disquisitiones Arithmeticae (which means Arithmetical Investigations). Carl Friedrich Gauss wrote it in Latin in 1798 when he was just 21 years old, and it was published in 1801.

In this book, Gauss brought together many ideas from earlier mathematicians like Fermat and Euler. He also added many important new discoveries of his own. Before Gauss, number theory was a collection of separate ideas. Gauss organized everything into a clear system, fixed old proofs, and expanded the subject in many ways. His work became the starting point for many mathematicians in the 1800s.

Dirichlet's Contributions

Peter Gustav Lejeune Dirichlet was a German mathematician who made big steps in the 1830s. He proved the first class number formula, which helps understand certain types of numbers. He also proved the Dirichlet unit theorem, which is a very important result about the "units" (numbers that divide 1) in algebraic number fields.

Dirichlet also helped prove cases of Fermat's Last Theorem for n = 5 and n = 14.

Dedekind and the Idea of Ideals

Richard Dedekind was inspired by Dirichlet's work. In 1863, he published Dirichlet's lectures on number theory. Later, in 1879 and 1894, Dedekind added new sections to the book that introduced the idea of an "ideal."

An ideal is a special collection of numbers within a ring (a type of number system). This concept was very important because it helped solve problems where numbers didn't break down into unique prime factors in the usual way. The idea of ideals was further developed by other mathematicians like David Hilbert and Emmy Noether.

Hilbert's Big Report

David Hilbert helped bring together the field of algebraic number theory with his 1897 book called Zahlbericht (meaning "report on numbers"). He also solved a major number theory problem called Waring's problem. Hilbert often used "existence proofs," which show that solutions must exist, even if they don't show how to find them.

Artin and Reciprocity

Emil Artin developed the Artin reciprocity law in the 1920s. This law is a very general theorem in number theory that connects different parts of the field. It generalized earlier "reciprocity laws" like the quadratic reciprocity law, which describes relationships between prime numbers.

Modern Breakthroughs

Around 1955, Japanese mathematicians Goro Shimura and Yutaka Taniyama noticed a possible connection between two very different areas of math: elliptic curves (special types of curves) and modular forms (special types of functions). This idea became the modularity theorem. It says that every elliptic curve is "modular," meaning it can be linked to a unique modular form.

This idea was first thought to be unlikely, but it gained importance when mathematician André Weil found evidence for it. The theorem became a key part of the Langlands program, a list of important math problems.

From 1993 to 1994, Andrew Wiles finally proved the modularity theorem for a certain type of elliptic curve. This proof, combined with another theorem, finally solved Fermat's Last Theorem! Many mathematicians had thought both theorems were impossible to prove. Wiles's proof was very complex and used many advanced techniques from different areas of mathematics.

Basic Ideas in Algebraic Number Theory

When Unique Factorization Fails

In regular whole numbers, we know that every number can be broken down into a unique set of prime numbers. For example, 12 = 2 × 2 × 3. This is called the fundamental theorem of arithmetic.

However, in some of the more complex number systems studied in algebraic number theory, this unique factorization doesn't always happen!

Let's look at an example in the number system called $Z [\sqrt -5]$ . In this system, numbers look like a + b√-5, where a and b are whole numbers. Consider the number 9. We can factor it in two different ways:

9 = 3 × 3
9 = (2 + √-5) × (2 - √-5)

In this system, 3, (2 + √-5), and (2 - √-5) are all "irreducible" numbers, meaning they can't be factored any further. But notice that 3 divides 9, but it doesn't divide (2 + √-5) or (2 - √-5). This means 3 is not a "prime" number in this system, even though it's irreducible. This shows that unique factorization (where factors are unique up to order and "units") doesn't hold here.

Factorization into Prime Ideals

To fix the problem of unique factorization, mathematicians came up with the idea of "ideals." Even if numbers themselves don't factor uniquely, their ideals do!

If you have an ideal (a special collection of numbers) in these number systems, it can always be broken down into a unique product of "prime ideals." This is a very powerful idea. It means that even when unique factorization of numbers fails, unique factorization of ideals always works.

Historically, this idea came from Ernst Kummer's "ideal numbers," which were used to try and prove Fermat's Last Theorem. Later, Dedekind developed the formal concept of ideals.

The Ideal Class Group

Unique factorization fails when there are prime ideals that can't be created by just one number. The "ideal class group" is a mathematical tool that measures exactly how much unique factorization fails in a number system.

The number of elements in the class group is called the class number. For example, the class number of $Q (\sqrt -5)$ (the number system we looked at earlier) is 2. This means there are only two "types" of ideals: those that can be generated by a single number, and those that cannot.

Units in Number Systems

In the system of whole numbers, the only "units" (numbers that divide 1) are 1 and -1. But in other number systems, there can be many more units.

For example, in the Gaussian integers (numbers like a + bi), the units are 1, -1, i, and -i. In some systems, like those involving square roots of positive numbers (e.g., $Z [\sqrt 3]$ ), there can be infinitely many units! For instance, every power of (2 + √3) is a unit in that system.

Dirichlet's unit theorem tells us exactly how many "independent" units there are in a given number system. It's a very important result that helps us understand the structure of these number systems.

Zeta Functions

The Dedekind zeta function is a mathematical function that helps describe how prime ideals behave in a number system. It's similar to the famous Riemann zeta function. These zeta functions are powerful tools for studying the properties of numbers and ideals.

Major Results in Algebraic Number Theory

Class Group is Finite

One of the most important results in algebraic number theory is that the ideal class group of any algebraic number field is always finite. This means that even if unique factorization fails, it doesn't fail in an infinite number of ways; there's a limited number of "types" of ideals. The size of this group is called the class number.

Dirichlet's Unit Theorem

As mentioned before, Dirichlet's unit theorem describes the structure of the "units" in a number system. It tells us that the group of units is made up of a finite part (the roots of unity, like i or -1) and a "free" part, which has a specific number of independent units. This number depends on how the number system can be "embedded" into real or complex numbers.

Reciprocity Laws

A reciprocity law is a generalization of the law of quadratic reciprocity. The quadratic reciprocity law tells us when one prime number is a "perfect square" when you divide by another prime number. For example, it helps us know if 5 is a square when you divide by 3 (it's not, since 1²=1, 2²=4, 3²=9=0 mod 3, 4²=16=1 mod 3, 5²=25=1 mod 3), or if 3 is a square when you divide by 5 (it's not, since 1²=1, 2²=4, 3²=9=4, 4²=16=1, 5²=25=0 mod 5).

Reciprocity laws extend this idea to higher powers and more complex number systems. They are deep connections between different prime numbers and their properties.

Class Number Formula

The class number formula is a powerful equation that connects the class number (the size of the ideal class group) with other important properties of a number system, including its Dedekind zeta function. It's a way to link different mathematical ideas together.

Related Areas

Algebraic number theory is connected to many other parts of mathematics. It uses tools from homological algebra and ideas from algebraic geometry (the study of geometric shapes using algebra). When mathematicians study higher-dimensional shapes using number systems, it's called arithmetic geometry. This field also helps in understanding certain complex geometric objects called arithmetic hyperbolic 3-manifolds.