A targeted Martinet search

Abstract

Abstract. Constructing number fields with prescribed ramification is an important problem in computational number theory. In this paper, we consider the problem of computing all imprimitive number fields of a given degree which are unramified outside of a given finite set of primes S by combining the techniques of targeted Hunter searches with Martinet's relative version of Hunter's theorem. We then carry out this algorithm to generate complete tables of imprimitive number fields for degrees 4 through 10 and certain sets S of small primes.An important problem in the study of fields is to determine all number fields of a fixed degree having a prescribed ramification structure. This paper will focus on finding all imprimitive number fields of a given degree unramified outside of a finite set of primes.Hunter's theorem has been used extensively for computing all primitive number fields of a given degree with absolute discriminant below a given bound. In [13], Martinet gives a version of Hunter's theorem suitable for relative extensions which has been used to carry out similar searches for imprimitive fields [4,5,16,17,18,19]. Note, however, that for even modest degree fields and small sets of primes, such as S = {2, 3}, using a standard Hunter search to find all fields unramified outside S can become computationally burdensome. This can be ameliorated by carrying out a targeted Hunter search where one searches for all fields with specific discriminants, but only those possible for fields unramified away from S. This approach was introduced in [6] and refined in [7] to determine all sextic and septic fields with S = {2, 3}, respectively. It has subsequently been used to investigate fields of degrees 8 and 9 ramified at a single prime in [11,12].In this paper, we combine Martinet's theorem with the targeted search technique to form what we call a targeted Martinet search. We then demonstrate the algorithm by using it to compute complete tables of imprimitive decic fields with prescribed ramification.Section 1 describes the process of conducting a number field search based on Martinet's theorem. The size of the relative extensions considered in applications here are larger than those in the literature. Section 2 describes how targeting can be used with a search described in Section 1, with details on combining congruences and archimedean bounds given in Section 3. Finally, Section 4 summarizes the results of several searches carried out using the methods in this paper.

NONSOLVABLE POLYNOMIALS WITH FIELD DISCRIMINANT 5A

“…Driver and Jones [8] have successfully specialized this family to get complete lists of quintics with certain prescribed ramification behavior over quadratic fields F . Our search is for a single field only, and so the complications of ensuring that a list is complete are not present.…”

“…To move past the projective level, it is best to first restate the projective level using the sextic polynomial (10) rather than the quintic polynomial (8). Just as each Fr Π is completely determined by a partition in S 5 which we now call λ 5,Π , so too each Fr Π is completely determined by a partition of six λ 6,Π .…”