A monic polynomial $f(x)\in {\mathbb Z}[x]$ of degree N is called monogenic if $f(x)$ is irreducible over ${\mathbb Q}$ and $\{1,\theta ,\theta ^2,\ldots ,\theta ^{N-1}\}$ is a basis for the ring of integers of ${\mathbb Q}(\theta )$, where $f(\theta )=0$. We use the classification of the Galois groups of quartic polynomials, due to Kappe and Warren [‘An elementary test for the Galois group of a quartic polynomial’, Amer. Math. Monthly 96(2) (1989), 133–137], to investigate the existence of infinite collections of monogenic quartic polynomials having a prescribed Galois group, such that each member of the collection generates a distinct quartic field. With the exception of the cyclic case, we provide such an infinite single-parameter collection for each possible Galois group. We believe these examples are new and we provide evidence to support this belief by showing that they are distinct from other infinite collections in the literature. Finally, we devote a separate section to the cyclic case.

A monic polynomial $f(x)\in {\mathbb Z}[x]$ of degree N is called monogenic if $f(x)$ is irreducible over ${\mathbb Q}$ and $\{1,\theta ,\theta ^2,\ldots ,\theta ^{N-1}\}$ is a basis for the ring of integers of ${\mathbb Q}(\theta )$, where $f(\theta )=0$. We prove that there exist exactly three distinct monogenic trinomials of the form $x^4+bx^2+d$ whose Galois group is the cyclic group of order 4. We also show that the situation is quite different when the Galois group is not cyclic.

The lifting problem for universal quadratic forms over a totally real number field K consists of determining the existence or otherwise of a quadratic form with integer coefficients (or $\mathbb {Z}$-form) that is universal over K. We prove the nonexistence of universal $\mathbb {Z}$-forms over simplest cubic fields for which the integer parameter is big enough. The monogenic case is already known. We prove the nonexistence in the nonmonogenic case by using the existence of a totally positive nonunit algebraic integer in K with minimal (codifferent) trace equal to one.

In this paper, we obtain a precise formula for the one-level density of L-functions attached to non-Galois cubic Dedekind zeta functions. We find a secondary term which is unique to this context, in the sense that no lower-order term of this shape has appeared in previously studied families. The presence of this new term allows us to deduce an omega result for cubic field counting functions, under the assumption of the Generalised Riemann Hypothesis. We also investigate the associated L-functions Ratios Conjecture and find that it does not predict this new lower-order term. Taking into account the secondary term in Roberts’s conjecture, we refine the Ratios Conjecture to one which captures this new term. Finally, we show that any improvement in the exponent of the error term of the recent Bhargava–Taniguchi–Thorne cubic field counting estimate would imply that the best possible error term in the refined Ratios Conjecture is $O_\varepsilon (X^{-\frac 13+\varepsilon })$ . This is in opposition with all previously studied families in which the expected error in the Ratios Conjecture prediction for the one-level density is $O_\varepsilon (X^{-\frac 12+\varepsilon })$ .

We prove that there is a positive proportion of L-functions associated to cubic characters over $\mathbb F_q[T]$ that do not vanish at the critical point $s=1/2$. This is achieved by computing the first mollified moment using techniques previously developed by the authors in their work on the first moment of cubic L-functions, and by obtaining a sharp upper bound for the second mollified moment, building on work of Lester and Radziwiłł, which in turn develops further ideas from the work of Soundararajan, Harper and Radziwiłł. We work in the non-Kummer setting when $q\equiv 2 \,(\mathrm {mod}\,3)$, but our results could be translated into the Kummer setting when $q\equiv 1\,(\mathrm {mod}\,3)$ as well as into the number-field case (assuming the generalised Riemann hypothesis). Our positive proportion of nonvanishing is explicit, but extremely small, due to the fact that the implied constant in the upper bound for the mollified second moment is very large.

We discuss the origin, an improved definition and the key reciprocity property of the trilinear symbol introduced by Rédei [16] in the study of 8-ranks of narrow class groups of quadratic number fields. It can be used to show that such 8-ranks are ‘governed’ by Frobenius conditions on the primes dividing the discriminant, a fact used in the recent work of A. Smith [18, 19]. In addition, we explain its impact in the progress towards proving my conjectural density for solvability of the negative Pell equation $x^2-dy^2=-1$ .

In this paper, we prove a one level density result for the low-lying zeros of quadratic Hecke L-functions of imaginary quadratic number fields of class number 1. As a corollary, we deduce, essentially, that at least $(19-\cot (1/4))/16 = 94.27\ldots \%$ of the L-functions under consideration do not vanish at 1/2.

The notion of $\theta $ -congruent numbers is a generalisation of congruent numbers where one considers triangles with an angle $\theta $ such that $\cos \theta $ is a rational number. In this paper we discuss a criterion for a natural number to be $\theta $ -congruent over certain real number fields.

In this paper we prove some one-level density results for the low-lying zeros of families of quadratic and quartic Hecke $L$-functions of the Gaussian field. As corollaries, we deduce that at least 94.27% and 5%, respectively, of the members of the quadratic family and the quartic family do not vanish at the central point.

A number field K with a ring of integers 𝒪K is called a Pólya field, if the 𝒪K-module of integer-valued polynomials on 𝒪K has a regular basis, or equivalently all its Bhargava factorial ideals are principal [1]. We generalize Leriche's criterion [8] for Pólya-ness of Galois closures of pure cubic fields, to general S3-extensions of ℚ. Also, we prove for a real (resp. imaginary) Pólya S3-extension L of ℚ, at most four (resp. three) primes can be ramified. Moreover, depending on the solvability of unit norm equation over the quadratic subfield of L, we determine when these sharp upper bounds can occur.

Starting with a paper of Jacobson from the 1960s, many authors became interested in characterizing all algebraic number fields in which each integer is the sum of pairwise distinct units. Although there exist many partial results for number fields of low degree, a full characterization of these number fields is still not available. Narkiewicz and Jarden posed an analogous question for sums of units that are not necessarily distinct. In this paper we propose a generalization of these problems. In particular, for a given rational integer n we consider the following problem. Characterize all number fields for which every integer is a linear combination of finitely many units εi in a way that the coefficients ai∈ℕ are bounded by n. The paper gives several partial results on this problem. In our proofs we exploit the fact that these representations are related to symmetric beta expansions with respect to Pisot bases.

In this paper, we prove some one level density results for the low-lying zeros of families of L-functions. More specifically, the families under consideration are that of L-functions of holomorphic Hecke eigenforms of level 1 and weight k twisted with quadratic Dirichlet characters and that of cubic and quartic Dirichlet L-functions.

We give an explicit treatment of cubic function fields of characteristic at least five. This includes an efficient technique for converting such a field into standard form, formulae for the field discriminant and the genus, simple necessary and sufficient criteria for non-singularity of the defining curve, and a characterization of all triangular integral bases. Our main result is a description of the signature of any rational place in a cubic extension that involves only the defining curve and the order of the base field. All these quantities only require simple polynomial arithmetic as well as a few square-free polynomial factorizations and, in some cases, square and cube root extraction modulo an irreducible polynomial. We also illustrate why and how signature computation plays an important role in computing the class number of the function field. This in turn has applications to the study of zeros of zeta functions of function fields.

We prove that there are only $O(H^{3+\epsilon})$ quartic integer polynomials with height at most $H$ and a Galois group which is a proper subgroup of $S_4$. This improves in the special case of degree four a bound by Gallagher that yielded $O(H^{7/2} \log H)$.

The number of cyclic cubic fields with a given conductor and a given index is determined.

It is shown that the Steinitz invariants of the cubic extensions of a number field are uniformly distributed in the class group when the cubic extensions are ordered by the ideal norm of their relative discriminants. This remains true even if the extensions are restricted by specifying their splitting type at a finite number of places. The same statement is also proved for quadratic extensions.

An asymptotic formula is obtained for the number of cyclic quartic fields over $Q$ with discriminant $\le \,x$.

Let K be the simplest cubic field defined by the irreducible polynomial

where m is a nonnegative rational integer such that m2 + 3m + 9 is square-free. We estimate the value of the Dedekind zeta function ζK(s) at s = −1 and get class number 1 criterion for the simplest cubic fields.

A formula is obtained for the rank of the 2-Sylow subgroup of the ideal class group of imaginary bicyclic biquadratic fields. This formula involves the number of primes that ramify in the field, the ranks of the 2-Sylow subgroups of the ideal class groups of the quadratic subfields and the rank of a Z2-matrix determined by Legendre symbols involving pairs of ramified primes. As applications, all subfields with both 2- class and class group Z2×Z2 are determined. The final results assume the completeness of D. A. Buell’s list of imaginary fields with small class numbers.

Let q = p if p is an odd prime, q = 4 if p = 2. Let ζq be any primitive q-th root of unity, and let . We study the family of polynomials where Rn(X) and Sn(X) are the polynomials in the expansion We show that for fixed n, Pn(X; a) is irreducible for all but finitely many a ∈ O, and for p = 3, we show that it is irreducible for all a ∈ O. The roots are all real and are permuted cyclically by a linear fractional transformation defined over the real pn-th cyclotomic field. From the roots we obtain a non-maximal set of independent units for the splitting field. In the last section we briefly treat extensions of our methods to composite p.