Atlas / Mathematics / The Number Theory Thread

Field · Emerged 1832 – 1871

Algebraic Number Theory

What happens to primes and factorisation when the whole numbers are extended to larger number systems?

5 chapters4 min read6 turning points 2 open problems

Branched from: Elementary Number Theory
Branched into: Abstract Algebra + Arithmetic Geometry
Figures: Carl Friedrich Gauss, Gabriel Lamé, Joseph Liouville, Ernst Kummer, Richard Dedekind, Teiji Takagi, Emil Artin, Kurt Heegner, Alan Baker, Harold Stark

In brief

Algebraic number theory studies number systems that extend the integers, such as numbers of the form $a + b\sqrt{-5}$ or $a + bi$ , built by adding roots of polynomial equations. Many problems about ordinary whole numbers become easier in these larger systems, with one catch: factorisation into primes may no longer be unique.

The field exists because of that catch. In 1847 an announced proof of Fermat's Last Theorem collapsed because it assumed unique factorisation where it fails. Repairing it created ideals, class groups and, eventually, class field theory, one of the great structures of twentieth-century mathematics.

Key ideas

Number field and algebraic integersEnters 1832: A number field is the rational numbers with finitely many roots of polynomials added, like $\mathbb{Q}(i)$ . Its algebraic integers play the role of whole numbers, like the Gaussian integers $a + bi$ .
Failure of unique factorisationEnters 1847: Among numbers $a + b\sqrt{-5}$ , $6 = 2 \cdot 3 = (1 + \sqrt{-5})(1 - \sqrt{-5})$ , two genuinely different factorisations into irreducible pieces.
IdealEnters 1871: A set of numbers closed under addition and under multiplication by anything in the system. Dedekind showed that ideals always factor uniquely into prime ideals, restoring what numbers had lost.
Class numberEnters 1844 – 1850: A number that measures how badly unique factorisation fails in a number system. It is 1 exactly when factorisation is unique.
Class field theoryEnters 1920 – 1927: A complete description of a large family of extensions of a number field, the abelian ones, in terms of arithmetic inside the field itself. It is the grand generalisation of quadratic reciprocity.

Chapter I

A New Kind of Integer

The whole numbers can be enlarged. Gauss, studying which numbers are fourth powers modulo a prime, worked in 1832 with numbers of the form $a + bi$ , where $i^2 = -1$ . These Gaussian integers have their own primes, and they factor into them uniquely, just as ordinary integers do. Inside this larger system old questions became easy. A prime like 5 splits as $(2 + i)(2 - i)$ , which is exactly why $5 = 2^2 + 1^2$ is a sum of two squares.

It was natural to try the same trick on Fermat's Last Theorem. The equation $x^n + y^n = z^n$ becomes a product once you allow complex $n$ th roots of unity: $x^n + y^n$ factors into $n$ linear pieces. If factorisation into primes were unique in that system, the theorem would follow.

Chapter II

The Proof That Failed

On 1 March 1847 Gabriel Lamé announced exactly that proof to the Paris Academy. Joseph Liouville rose at once to ask the obvious question: is factorisation unique there? Augustin Cauchy, meanwhile, claimed to have a proof of his own. Within weeks the question was answered from Breslau. Ernst Kummer had shown three years earlier that unique factorisation fails for some exponents, and he already had a way around it.

The failure is easy to see in a simpler system. Among numbers $a + b\sqrt{-5}$ ,

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

and none of the four factors can be broken down further. Arithmetic had lost its most basic law. Few fields can point to the moment they were forced into existence so precisely: it was this collapse that made algebraic number theory a subject of its own.

Chapter III

Ideal Numbers, Then Ideals

Kummer's remedy was audacious. He invented ideal numbers: phantom factors, not in the system at all, that behaved like primes and restored unique factorisation. With them he proved Fermat's Last Theorem for all "regular" primes, the first proof to cover a whole class of exponents at once.

In 1871 Richard Dedekind made the phantoms concrete. An ideal is a set of actual numbers, closed under addition and under multiplication by anything in the system. Ideals, unlike numbers, always factor uniquely into prime ideals, in every number field. The shift of attention from individual numbers to sets with structure became the template for modern abstract algebra, as Emmy Noether, working on Dedekind's foundations, later made explicit.

Chapter IV

A Closer Look: Primes That Split, and Ideals That Repair

In the Gaussian integers $a + bi$ , some ordinary primes stop being prime. For example

5 = (2 + i)(2 - i), \qquad 13 = (3 + 2i)(3 - 2i),

while 3, 7 and 11 cannot be factored. The rule is exact: an odd prime splits precisely when it leaves remainder 1 on division by 4. Splitting means $p = (a + bi)(a - bi) = a^2 + b^2$ , so this is Fermat's theorem on sums of two squares, $5 = 2^2 + 1^2$ and $13 = 3^2 + 2^2$ , recovered as a fact about factorisation in a larger number system.

In the numbers $a + b\sqrt{-5}$ unique factorisation fails:

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

and none of these four factors can be broken down further. Dedekind's ideals repair the damage by factoring more finely. Let $P$ be the ideal generated by $2$ and $1 + \sqrt{-5}$ , and $Q$ and $Q'$ the ideals generated by $3$ with $1 + \sqrt{-5}$ and with $1 - \sqrt{-5}$ . None of these ideals comes from a single number, which is exactly why no element can play their role. But they multiply out to

(2) = P^2, \quad (3) = Q\,Q', \quad (1 + \sqrt{-5}) = P\,Q, \quad (1 - \sqrt{-5}) = P\,Q' ,

so both factorisations of 6 are the same prime-ideal factorisation, $(6) = P^2 Q Q'$ , grouped in two different ways. The class number measures how many such "phantom" factors a number system needs. For $a + b\sqrt{-5}$ it is 2, meaning factorisation fails in the mildest possible way.

Chapter V

Reciprocity, Completed and Extended

Gauss's quadratic reciprocity was the first of a family of laws about which numbers are powers modulo primes. Hilbert conjectured their general form. Teiji Takagi proved it in 1920 and Emil Artin crowned it in 1927. Class field theory describes every abelian extension of a number field from arithmetic inside the field. Old problems were closed too. Gauss's question of which imaginary quadratic systems factor uniquely was answered, first by Kurt Heegner, whose correct proof was ignored for fifteen years.

The next step, extending class field theory beyond the abelian case, is the Langlands program, the largest open project in number theory. Its first great success, the modularity of elliptic curves, required uniting this field with algebraic geometry. That union is arithmetic geometry.

Turning points6

1832REFORMULATION
Gauss introduces the Gaussian integers
To state the reciprocity law for fourth powers, Gauss works with numbers $a + bi$ , where $i^2 = -1$ . He shows that they factor uniquely into primes, just as ordinary integers do. Number theory gains its first "larger" number system, and questions about ordinary primes, such as which are sums of two squares, become transparent inside it.
Carl Friedrich Gauss
› Sources (1)
- Gauss, C. F. (1832). Theoria residuorum biquadraticorum. Commentatio secunda. Commentationes Societatis Regiae Scientiarum Gottingensis Recentiores 7.
1847CRISIS
Lamé's proof of Fermat's Last Theorem collapses
In March 1847 Gabriel Lamé announces a proof of Fermat's Last Theorem to the Paris Academy. It factors $x^n + y^n$ using complex roots of unity. Joseph Liouville immediately asks whether factorisation into primes is unique in those number systems. Within weeks, a letter from Ernst Kummer reveals that he had shown three years earlier that it is not. The proof, and a rival one Cauchy was preparing, were dead.
Gabriel Lamé, Joseph Liouville, Ernst Kummer
› Sources (1)
- Edwards, H. M. (1977). Fermat's Last Theorem: A Genetic Introduction to Algebraic Number Theory. Springer.
1844 – 1850REFORMULATION
Kummer's ideal numbers
Ernst Kummer restores unique factorisation by inventing "ideal numbers", phantom factors that behave like primes even though they are not numbers in the system. With them he proves Fermat's Last Theorem for all "regular" prime exponents, which includes every odd prime below 37. It was the first proof to cover a whole class of exponents at once.
Ernst Kummer
› Sources (1)
- Kummer, E. E. (1847). Zur Theorie der complexen Zahlen. Journal für die reine und angewandte Mathematik 35: 319–326.
1871REFORMULATION
Dedekind replaces ideal numbers with ideals
In a supplement to Dirichlet's lectures on number theory, Richard Dedekind makes Kummer's phantoms concrete. An ideal is a set of actual numbers, and every ideal factors uniquely into prime ideals in any number field. The move from individual elements to sets with structure became a model for all of modern algebra.
Richard Dedekind
› Sources (1)
- Dedekind, R. (1871). Supplement X in P. G. L. Dirichlet, Vorlesungen über Zahlentheorie (2nd ed.). Vieweg, Braunschweig.
1920 – 1927PROOF
Class field theory is completed
Building on Hilbert's conjectures, Teiji Takagi (1920) proves the main theorems describing all abelian extensions of a number field, and Emil Artin (1927) proves the general reciprocity law that sums them up. Every reciprocity law since Gauss becomes a special case of one theorem.
Teiji Takagi, Emil Artin
› Sources (2)
- Takagi, T. (1920). Über eine Theorie des relativ Abel'schen Zahlkörpers. Journal of the College of Science, Imperial University of Tokyo 41: 1–133.
- Artin, E. (1927). Beweis des allgemeinen Reziprozitätsgesetzes. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 5: 353–363.
1952 – 1967PROOFContested
Gauss's class number one problem is solved
Gauss had listed nine imaginary quadratic number systems with unique factorisation and asked whether there were others. There are not. Alan Baker (1966) and Harold Stark (1967) proved it by different methods, completing a question left open since 1801.
What is disputed
Kurt Heegner, a German private scholar outside the universities, had published a proof in 1952 using modular functions. It was generally believed to contain gaps and was ignored. After Baker and Stark's work, Stark and others re-examined it and concluded that Heegner's proof was essentially correct, with a minor gap that was easily filled. Heegner had died in 1965, unrecognised. He is now usually credited as the first to solve the problem.
Kurt Heegner, Alan Baker, Harold Stark
› Sources (2)
- Heegner, K. (1952). Diophantische Analysis und Modulfunktionen. Mathematische Zeitschrift 56(3): 227–253.
- Goldfeld, D. (1985). Gauss's class number problem for imaginary quadratic fields. Bulletin of the AMS 13(1): 23–37.

Applications

Where it is used

Cryptography
Post-quantum encryption over number rings
The new encryption standards meant to survive quantum computers do their arithmetic in rings of algebraic integers, typically cyclotomic rings. Their security rests on hard problems about lattices built from those rings.
› Sources (1)
- Lyubashevsky, V., Peikert, C. & Regev, O. (2010). On ideal lattices and learning with errors over rings. In Advances in Cryptology — EUROCRYPT 2010, Lecture Notes in Computer Science 6110: 1–23.

Open problems

Where the map runs out

Conjectured, unproven

The Langlands program

Proved in many special cases; the general conjectures remain open as of 2026.

In a 1967 letter to André Weil, Robert Langlands proposed a web of conjectures linking number theory (Galois groups, which describe symmetries of solutions to polynomial equations) with analysis (automorphic forms, highly symmetric functions). It would be class field theory extended to all extensions, not only the abelian ones.

Why it is hard: The conjectures connect objects from fields that developed separately and speak different languages. Each proved case, such as the modularity of elliptic curves behind Fermat's Last Theorem, has required years of new mathematics. The general case is widely regarded as one of the largest programmes in modern mathematics.
What resolving it unlocks: A "grand unified theory" of number theory, in which reciprocity laws, L-functions and the symmetries of equations become aspects of one structure.

› Sources (2)

Langlands, R. P. (1970). Problems in the theory of automorphic forms. In Lectures in Modern Analysis and Applications III, Lecture Notes in Mathematics 170: 18–61. Springer.
Frenkel, E. (2013). Love and Math: The Heart of Hidden Reality. Basic Books.

Conjectured, unproven

Infinitely many real quadratic fields with unique factorisation?

Conjectured by Gauss; open as of 2026.

For number systems like $a + b\sqrt{d}$ with $d$ positive, factorisation seems to be unique surprisingly often. Gauss conjectured that it happens for infinitely many $d$ . The Cohen–Lenstra heuristics, backed by numerical evidence, predict that about three-quarters of prime $d$ qualify, but it has not been proved that even infinitely many do.

Why it is hard: In real quadratic fields the class number is entangled with the size of the "fundamental unit", a quantity that fluctuates wildly and is hard to control. Methods that settled the imaginary case do not apply.
What resolving it unlocks: It would confirm the statistical picture of class groups behind the Cohen–Lenstra heuristics, now used widely in arithmetic statistics.

› Sources (1)

Cohen, H. (1993). A Course in Computational Algebraic Number Theory. Springer.