Minimum discriminants of imprimitive decic fields.

*(English)*Zbl 1298.11120A permutation group \(G\) acting on a set \(X\) is called imprimitive if \(G\) is transitive and \(G\) does preserve a nontrivial partition. A decic field (shortly decic) is a field of absolute degree \(10\).

The authors compute the minimum discriminants of imprimitive decics for different combinations of Galois group and signature. The article is essentially of computational nature with use of PARI/GP, the Bordeaux megrez tables of number fields [“The Bordeaux Database of Number Fields”, ftp://megrez.math.u-bordeaux. fr/pub/numberfields], J. Jones and D. Roberts table of number fields [“Tables of number fields with prescribed ramification”, http://math.la.asu.edu/~jj/numberfields] and the BOINC system [BOINC, “Berkeley Open Infrastructure for Network Computing”, http://boinc. berkeley.edu]. The results are given on the form of tables containing minimal polynomials and discriminants.

Let \(L\) be a number field of a degree \(n\), \(D_L\) its discriminant, and \(d_L := |D_L|\). Let \(G(L)\) be a transitive subgroup of the permutation group \(S_n\) giving the Galois group of the Galois closure of \(L/\mathbb Q\), which is well-defined up to conjugation. Let \(r_1\) denote the number of real places of \(L\).

For a given positive integer n and group \(G\leq S_n\), several authors have contributed to the knowledge of minimizing \(d_L\) for fields \(L\) with \([L : \mathbb Q] = n\) and \(G(L) = G,\) in particular J. Klüners and G. Malle [LMS J. Comput. Math. 4, 182–196 (2001; Zbl 1067.11516)] have results primarily for degrees \(\leq 8\), and some results in degree \(9\).

The authors consider imprimitive decic and attempt to determine minimal examples for fixed \(G\) and \(r_1\).

The computation for imprimitive decics divides into three cases: one refer to decics \(L\) which contain a quadratic but not quintic subfield as Type 2 fields, those which contain a quintic but no quadratic subfield as Type 5 fields, and those which contain both as Type 2–5 fields. For Type 2 fields, the author employs computer searches based on Martinet’s generalization of Hunter’s theorem see [H. Cohen, Advanced topics in computational number theory. Graduate Texts in Mathematics. 193. New York, NY: Springer. (2000; Zbl 0977.11056)]. Some of these were standard Martinet searches (based on the approach of geometry of numbers), and some were targeted Martinet searches in the sense of [E. D. Driver and J. W. Jones, Math. Comput. 78, No. 266, 1109–1117 (2009; Zbl 1213.11202)]; this is explained in Section 1 of the paper.

Section 2 treats Type 5 decics using class field theory. These computations rely on having a sufficiently large complete list of quintic fields to work from. The authors have extended the range of known totally real quintic fields \(K\) from \(d_K \leq 2 \cdot 10^7\) to \(d_K \leq 10^8\) which in turn, allowed them to determine more minimal decic discriminants.

In Section 3, they describe results for the simplest case, Type 2–5 decic fields. These fields are tensor products of their subfields, so it suffices to work with tables of lower degree fields, and the extended list of totally real quintic fields obtained here.

The authors compute the minimum discriminants of imprimitive decics for different combinations of Galois group and signature. The article is essentially of computational nature with use of PARI/GP, the Bordeaux megrez tables of number fields [“The Bordeaux Database of Number Fields”, ftp://megrez.math.u-bordeaux. fr/pub/numberfields], J. Jones and D. Roberts table of number fields [“Tables of number fields with prescribed ramification”, http://math.la.asu.edu/~jj/numberfields] and the BOINC system [BOINC, “Berkeley Open Infrastructure for Network Computing”, http://boinc. berkeley.edu]. The results are given on the form of tables containing minimal polynomials and discriminants.

Let \(L\) be a number field of a degree \(n\), \(D_L\) its discriminant, and \(d_L := |D_L|\). Let \(G(L)\) be a transitive subgroup of the permutation group \(S_n\) giving the Galois group of the Galois closure of \(L/\mathbb Q\), which is well-defined up to conjugation. Let \(r_1\) denote the number of real places of \(L\).

For a given positive integer n and group \(G\leq S_n\), several authors have contributed to the knowledge of minimizing \(d_L\) for fields \(L\) with \([L : \mathbb Q] = n\) and \(G(L) = G,\) in particular J. Klüners and G. Malle [LMS J. Comput. Math. 4, 182–196 (2001; Zbl 1067.11516)] have results primarily for degrees \(\leq 8\), and some results in degree \(9\).

The authors consider imprimitive decic and attempt to determine minimal examples for fixed \(G\) and \(r_1\).

The computation for imprimitive decics divides into three cases: one refer to decics \(L\) which contain a quadratic but not quintic subfield as Type 2 fields, those which contain a quintic but no quadratic subfield as Type 5 fields, and those which contain both as Type 2–5 fields. For Type 2 fields, the author employs computer searches based on Martinet’s generalization of Hunter’s theorem see [H. Cohen, Advanced topics in computational number theory. Graduate Texts in Mathematics. 193. New York, NY: Springer. (2000; Zbl 0977.11056)]. Some of these were standard Martinet searches (based on the approach of geometry of numbers), and some were targeted Martinet searches in the sense of [E. D. Driver and J. W. Jones, Math. Comput. 78, No. 266, 1109–1117 (2009; Zbl 1213.11202)]; this is explained in Section 1 of the paper.

Section 2 treats Type 5 decics using class field theory. These computations rely on having a sufficiently large complete list of quintic fields to work from. The authors have extended the range of known totally real quintic fields \(K\) from \(d_K \leq 2 \cdot 10^7\) to \(d_K \leq 10^8\) which in turn, allowed them to determine more minimal decic discriminants.

In Section 3, they describe results for the simplest case, Type 2–5 decic fields. These fields are tensor products of their subfields, so it suffices to work with tables of lower degree fields, and the extended list of totally real quintic fields obtained here.

Reviewer: Roland Quême (Brax)

##### MSC:

11Y40 | Algebraic number theory computations |

11R21 | Other number fields |

11R29 | Class numbers, class groups, discriminants |

11R37 | Class field theory |

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\textit{E. D. Driver} and \textit{J. W. Jones}, Exp. Math. 19, No. 4, 475--479 (2010; Zbl 1298.11120)

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##### References:

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[3] | DOI: 10.1090/S0025-5718-08-02178-9 · Zbl 1213.11202 |

[4] | Klüners J., LMS J. Comput. Math. 4 pp 182– (2001) |

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