In this paper we describe how to compute smallest monic polynomials that define a given number field $\mathbb{K}$. We make use of the one-to-one correspondence between monic defining polynomials of $\mathbb{K}$ and algebraic integers that generate $\mathbb{K}$. Thus, a smallest polynomial corresponds to a vector in the lattice of integers of $\mathbb{K}$ and this vector is short in some sense. The main idea is to consider weighted coordinates for the vectors of the lattice of integers of $\mathbb{K}$. This allows us to find the desired polynomial by enumerating short vectors in these weighted lattices. In the context of the subexponential algorithm of Biasse and Fieker for computing class groups, this algorithm can be used as a precomputation step that speeds up the rest of the computation. It also widens the applicability of their faster conditional method, which requires a defining polynomial of small height, to a much larger set of number field descriptions.

In previous work, Ohno conjectured, and Nakagawa proved, relations between the counting functions of certain cubic fields. These relations may be viewed as complements to the Scholz reflection principle, and Ohno and Nakagawa deduced them as consequences of ‘extra functional equations’ involving the Shintani zeta functions associated to the prehomogeneous vector space of binary cubic forms. In the present paper, we generalize their result by proving a similar identity relating certain degree-$\ell$ fields to Galois groups $D_{\ell }$ and $F_{\ell }$, respectively, for any odd prime $\ell$; in particular, we give another proof of the Ohno–Nakagawa relation without appealing to binary cubic forms.

The discriminant of a trinomial of the form $x^{n}\pm \,x^{m}\pm \,1$ has the form $\pm n^{n}\pm (n-m)^{n-m}m^{m}$ if $n$ and $m$ are relatively prime. We investigate when these discriminants have nontrivial square factors. We explain various unlikely-seeming parametric families of square factors of these discriminant values: for example, when $n$ is congruent to 2 (mod 6) we have that $((n^{2}-n+1)/3)^{2}$ always divides $n^{n}-(n-1)^{n-1}$. In addition, we discover many other square factors of these discriminants that do not fit into these parametric families. The set of primes whose squares can divide these sporadic values as $n$ varies seems to be independent of $m$, and this set can be seen as a generalization of the Wieferich primes, those primes $p$ such that $2^{p}$ is congruent to 2 (mod $p^{2}$). We provide heuristics for the density of these sporadic primes and the density of squarefree values of these trinomial discriminants.

We formulate a conjecture which generalizes Darmon’s ‘refined class number formula’. We discuss relations between our conjecture and the equivariant leading term conjecture of Burns. As an application, we give another proof of the ‘except $2$-part’ of Darmon’s conjecture, which was first proved by Mazur and Rubin.

Gross and Rubin have made conjectures about special values of equivariant L-functions associated to abelian extensions of global fields. We describe a common refinement, due to Burns, and give evidence in favour of this conjecture for quadratic extensions and cyclotomic fields. We also note that the statement provides a new interpretation of further conjectures of Darmon and Gross.