The index of a quartic field defined by a trinomial X4 + aX + b

Consider an irreducible quartic polynomial of the form [Formula: see text], where [Formula: see text] satisfy [Formula: see text] or [Formula: see text], where [Formula: see text] denotes the exact power of a rational prime [Formula: see text] that divides an integer. Such a polynomial is called a [Formula: see text]-minimal quartic. Let [Formula: see text] be a field defined by a [Formula: see text]-minimal quartic. In this paper, we use [Formula: see text]-integral bases and introduce the concept of [Formula: see text]-index forms in order to determine the field index of [Formula: see text] via the coefficients of a defining polynomial in terms of certain congruence conditions.

Cohomological operations for the Brauer group and Witt kernels of a quartic field extension

Let [Formula: see text] be a field, [Formula: see text], [Formula: see text][Formula: see text], [Formula: see text] a quartic field extension. We investigate the divided power operation [Formula: see text] on the group [Formula: see text]. In particular, we show that any element of [Formula: see text] is a symbol [Formula: see text], where [Formula: see text], [Formula: see text], and [Formula: see text] is the quadratic trace form associated with the extension [Formula: see text]. As an application, we obtain certain results on the Stifel–Whitney maps [Formula: see text].

NEWTON POLYGONS AND p-INTEGRAL BASES OF QUARTIC NUMBER FIELDS

Let p be a prime number. In this paper we use an old technique of Ø. Ore, based on Newton polygons, to construct in an efficient way p-integral bases of number fields defined by p-regular equations. To illustrate the potential applications of this construction, we derive from this result an explicit description of a p-integral basis of an arbitrary quartic field in terms of a defining equation.

EXCEPTIONAL ELLIPTIC CURVES OVER QUARTIC FIELDS

We study the number of elliptic curves, up to isomorphism, over a fixed quartic field K having a prescribed torsion group T as a subgroup. Let T = ℤ/mℤ⊕ℤ/nℤ, where m|n, be a torsion group such that the modular curve X1(m, n) is an elliptic curve. Let K be a number field such that there is a positive and finite number of elliptic curves E over K having T as a subgroup. We call such pairs (T, K)exceptional. It is known that there are only finitely many exceptional pairs when K varies through all quadratic or cubic fields. We prove that when K varies through all quartic fields, there exist infinitely many exceptional pairs when T = ℤ/14ℤ or ℤ/15ℤ and finitely many otherwise.

On the Lagrange resolvents of a dihedral quintic polynomial

The cyclic quartic field generated by the fifth powers of the Lagrange resolvents of a dihedral quintic polynomialf(x)is explicitly determined in terms of a generator for the quadratic subfield of the splitting field off(x).