Section: Scientific Foundations

Number fields, class groups and other invariants

Participants : Bill Allombert, Athanasios Angelakis, Karim Belabas, Julio Brau, Jean-Paul Cerri, Henri Cohen, Jean-Marc Couveignes, Andreas Enge, Pierre Lezowski, Nicolas Mascot, Aurel Page.

Modern number theory has been introduced in the second half of the 19th century by Dedekind, Kummer, Kronecker, Weber and others, motivated by Fermat's conjecture: There is no non-trivial solution in integers to the equation $x^n+y^n=z^n$ for $n⩾3$. For recent textbooks, see [7] . Kummer's idea for solving Fermat's problem was to rewrite the equation as $(x+y)(x+\zeta y)(x+{\zeta }^{2}y)\cdots (x+{\zeta }^{n-1}y)=z^n$ for a primitive $n$-th root of unity $\zeta$, which seems to imply that each factor on the left hand side is an $n$-th power, from which a contradiction can be derived.

The solution requires to augment the integers by algebraic numbers, that are roots of polynomials in $\mathbb{Z}\left[X\right]$. For instance, $\zeta$ is a root of $X^n-1$, $\sqrt[3]{2}$ is a root of $X^3-2$ and $\frac{\sqrt{3}}{5}$ is a root of $25X^2-3$. A number field consists of the rationals to which have been added finitely many algebraic numbers together with their sums, differences, products and quotients. It turns out that actually one generator suffices, and any number field $K$ is isomorphic to $\mathbb{Q}\left[X\right]/\left(f\right(X\left)\right)$, where $f\left(X\right)$ is the minimal polynomial of the generator. Of special interest are algebraic integers, “numbers without denominators”, that are roots of a monic polynomial. For instance, $\zeta$ and $\sqrt[3]{2}$ are integers, while $\frac{\sqrt{3}}{5}$ is not. The ring of integers of $K$ is denoted by ${𝒪}_K$; it plays the same role in $K$ as $\mathbb{Z}$ in $\mathbb{Q}$.

Unfortunately, elements in ${𝒪}_K$ may factor in different ways, which invalidates Kummer's argumentation. Unique factorisation may be recovered by switching to ideals, subsets of ${𝒪}_K$ that are closed under addition and under multiplication by elements of ${𝒪}_K$. In $\mathbb{Z}$, for instance, any ideal is principal, that is, generated by one element, so that ideals and numbers are essentially the same. In particular, the unique factorisation of ideals then implies the unique factorisation of numbers. In general, this is not the case, and the class group ${Cl}_K$ of ideals of ${𝒪}_K$ modulo principal ideals and its class number $h_K=\left|{Cl}_K\right|$ measure how far ${𝒪}_K$ is from behaving like $\mathbb{Z}$.

Using ideals introduces the additional difficulty of having to deal with $\mathrm{𝑢𝑛𝑖𝑡𝑠}$, the invertible elements of ${𝒪}_K$: Even when $h_K=1$, a factorisation of ideals does not immediately yield a factorisation of numbers, since ideal generators are only defined up to units. For instance, the ideal factorisation $\left(6\right)=\left(2\right)·\left(3\right)$ corresponds to the two factorisations $6=2·3$ and $6=(-2)·(-3)$. While in $\mathbb{Z}$, the only units are 1 and $-1$, the unit structure in general is that of a finitely generated $\mathbb{Z}$-module, whose generators are the fundamental units. The regulator $R_K$ measures the “size” of the fundamental units as the volume of an associated lattice.

One of the main concerns of algorithmic algebraic number theory is to explicitly compute these invariants (${Cl}_K$ and $h_K$, fundamental units and $R_K$), as well as to provide the data allowing to efficiently compute with numbers and ideals of ${𝒪}_K$; see [31] for a recent account.

The analytic class number formula links the invariants $h_K$ and $R_K$ (unfortunately, only their product) to the $\zeta$-function of $K$, ${\zeta }_K\left(s\right):={\prod }_{𝔭\text{prime}\text{ideal}\text{of}{𝒪}_K}{\left(1-N{𝔭}^{-s}\right)}^{-1}$, which is meaningful when $\Re \left(s\right)>1$, but which may be extended to arbitrary complex $s\ne 1$. Introducing characters on the class group yields a generalisation of $\zeta$- to $L$-functions. The generalised Riemann hypothesis (GRH), which remains unproved even over the rationals, states that any such $L$-function does not vanish in the right half-plane $\Re \left(s\right)>1/2$. The validity of the GRH has a dramatic impact on the performance of number theoretic algorithms. For instance, under GRH, the class group admits a system of generators of polynomial size; without GRH, only exponential bounds are known. Consequently, an algorithm to compute ${Cl}_K$ via generators and relations (currently the only viable practical approach) either has to assume that GRH is true or immediately becomes exponential.

When $h_K=1$ the number field $K$ may be norm-Euclidean, endowing ${𝒪}_K$ with a Euclidean division algorithm. This question leads to the notions of the Euclidean minimum and spectrum of $K$, and another task in algorithmic number theory is to compute explicitly this minimum and the upper part of this spectrum, yielding for instance generalised Euclidean gcd algorithms.