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.
In this paper, we compute the minimum discriminants of imprimitive degree-10 fields for different combinations of Galois group and signature. We use class field theory when there is a quintic subfield, and a Martinet search in the more difficult case in which there is only a quadratic subfield.
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