We survey methods to compute three-point branched covers of the projective line, also known as Belyĭ maps. These methods include a direct approach, involving the solution of a system of polynomial equations, as well as complex analytic methods, modular forms methods, and $p$-adic methods. Along the way, we pose several questions and provide numerous examples.

Let $\mathbb{Q}\left(\alpha \right)$ be an algebraic number field given by the minimal polynomial $f$ of $\alpha$. We want to determine all subfields $\mathbb{Q}\left(\beta \right)\subset \mathbb{Q}\left(\alpha \right)$ of given degree. It is convenient to describe each subfield by a pair $(g,h)\in \mathbb{Z}\left[t\right]\times \mathbb{Q}\left[t\right]$ such that $g$ is the minimal polynomial of $\beta =h\left(\alpha \right)$. There is a bijection between the block systems of the Galois group of $f$ and the subfields of $\mathbb{Q}\left(\alpha \right)$. These block systems are computed using cyclic subgroups of the Galois group which we get from the Dedekind criterion. When a block system is known we compute the corresponding...

We generalize the concept of reduced Arakelov divisors and define C-reduced divisors for a given number C ≥ 1. These C-reduced divisors have remarkable properties, similar to the properties of reduced ones. We describe an algorithm to test whether an Arakelov divisor of a real quadratic field F is C-reduced in time polynomial in $log|{\Delta }_F|$ with ${\Delta }_F$ the discriminant of F. Moreover, we give an example of a cubic field for which our algorithm does not work.

Let $L=K\left(\alpha \right)$ be an extension of a number field $K$, where $\alpha$ satisfies the monic irreducible polynomial $P\left(X\right)=X^p-\beta$ of prime degree belonging to ${𝔬}_K\left[X\right]$ (${𝔬}_K$ is the ring of integers of $K$). The purpose of this paper is to study the monogenity of $L$ over $K$ by a simple and practical version of Dedekind’s criterion characterizing the existence of power integral bases over an arbitrary Dedekind ring by using the Gauss valuation and the index ideal. As an illustration, we determine an integral basis of a pure nonic field $L$ with a...

The class numbers h⁺ of the real cyclotomic fields are very hard to compute. Methods based on discriminant bounds become useless as the conductor of the field grows, and methods employing Leopoldt's decomposition of the class number become hard to use when the field extension is not cyclic of prime power. This is why other methods have been developed, which approach the problem from different angles. In this paper we extend one of these methods that was designed for real cyclotomic fields of prime...

We present algorithms for the computation of extreme binary Humbert forms in real quadratic number fields. With these algorithms we are able to compute extreme Humbert forms for the number fields $\mathbb{Q}\left(\sqrt{13}\right)$ and $\mathbb{Q}\left(\sqrt{17}\right)$. Finally we compute the Hermite-Humbert constant for the number field $\mathbb{Q}\left(\sqrt{13}\right)$.

We describe an algorithm due to Gauss, Shanks and Lagarias that, given a non-square integer $D\equiv 0,1$ mod $4$ and the factorization of $D$, computes the structure of the $2$-Sylow subgroup of the class group of the quadratic order of discriminant $D$ in random polynomial time in $log\left|D\right|$.