Algebraic Number Theory

Algebraic number theory is a branch of mathematics. It deals with integers, that is, with 1, 2, 3 and their negative counterparts. Above all, it concerns questions of divisibility: which number can be divided by which other number without remainder? The field’s special trick lies in enlarging the number range. One does not calculate only with ordinary integers, but adds further numbers, such as the square root of 2 or the square root of minus 1. In these extended number ranges, many problems that previously looked hopeless suddenly become solvable.

From Fermat’s Riddle to Cryptography

The best-known impetus for this field was a riddle from the 17th century. Pierre de Fermat claimed that the equation a to the power of n plus b to the power of n equals c to the power of n has no solution in integers as soon as n is greater than 2. For n equal to 2 there are solutions, such as 3, 4, and 5. For higher exponents, no one found a proof for over 350 years. In the search for one, mathematicians developed exactly the tools that today make up algebraic number theory.

The field became practically significant with the internet. When you open a page with a lock symbol in your browser, encryption runs in the background. Its security rests on the fact that very large numbers are extremely hard to break down into their prime factors. Prime factors are the indivisible building blocks of a number; for 12, these are 2, 2, and 3. How hard this decomposition really is, is answered by number theory.

The reverse direction is also a topic. The fastest known methods for factoring large numbers likewise come from this field. The so-called number field sieve directly exploits extended number ranges. Anyone who wants to build encryption must therefore know how far the attacks already reach.

What Happens When You Add Numbers

In the ordinary integers, a very useful law holds. Every number can be broken down into prime factors in exactly one way. 60 is always 2 times 2 times 3 times 5; there is no other decomposition. This uniqueness is the ground on which almost all of school mathematics stands.

If you extend the number range, this law can break down. If, for example, you add the square root of minus 5, the number 6 can be split into indecomposable factors in two different ways. This was a shock to 19th-century mathematics. Ernst Eduard Kummer saved the situation with an idea that can be thought of as invisible numbers. He introduced so-called ideals, in a sense building blocks beneath the visible numbers.

With these ideals, uniqueness is restored. One then no longer decomposes numbers but ideals, and that works uniquely again. How much a number range deviates from the ideal case is measured by a figure called the class number. If it is 1, the range behaves nicely, just like the familiar integers. If it is larger, things get complicated, and it is precisely these cases that are mathematically most interesting.

Traces in Technology and Headlines

In everyday life, you encounter this field mostly hidden. Every online payment, every messenger message, and every digital signature relies on number-theoretic methods. Error correction, too, uses related ideas, for instance when a scratched QR code can still be read. You notice none of this because the mathematics does its work invisibly.

In the news, this field regularly comes up in connection with quantum computers. A sufficiently large quantum computer could quickly factor large numbers and thereby break today’s encryption. That is why agencies and companies are working on post-quantum cryptography, that is, on methods that remain secure even then. These new methods often rely on lattices, another concept from number theory.

A common misconception is that algebraic number theory is the same as the algebra taught in school. There, one solves equations with x and y in the real numbers. Here, the only interest is in what is possible in integers, which is considerably harder. The field also comes up in AI topics: researchers like to test proof assistants and language models on number-theoretic problems, because the tasks are short to state but hard to solve.