Minimal Discriminants

Results known to date. One goal of our database is to provide fields with small (absolute value of the) discriminant for each Galois group and signature. In small degrees it is even possible to determine the field(s) with smallest discriminant. Let's comment on the present state of knowledge in this area (which is restricted to degrees less than 10).

It is very easy to enumerate the discriminants of quadratic fields.


Degree 3:

Belabas [Bel] gives a very efficient algorithm to enumerate cubic number fields.


Degree 4:

In [BFP] all quartic fields with absolute discriminant smaller than 106 are enumerated.


Degree 5:

There are huge tables of the smallest quintic fields due to [SPD]. These tables are sufficient to extract the smallest discriminants for all Galois groups and classes of involutions for degree 5.


Degree 6:

The general enumeration methods are not powerful enough to give the minima for all Galois groups in degree 6. The minimal discriminants for all signatures of degree 6 are computed in [Poh2]. [Oli2, FoPo1, FoPo2, FPD] have finished the computation of minimal discriminants of all signatures and all primitive Galois groups of degree 6. [BMO, Oli1] compute the minimal fields for imprimitive groups of degree 6. This yields enough information to determine the minimal fields for all groups and all conjugacy classes of that degree.

  • Imprimitive fields: [BMO, Oli1]

  • Primitive fields: [Oli2, FoPo1, FoPo2, FPD, Poh2]


Degree 7:

In degree 7 the minimal fields of each signature are known due to [Dia1, Dia2, Poh1].

This covers all signatures of the symmetric groups.

  • Symmetric group: [Dia1, Dia2, Poh1]

  • Other non-solvable groups: [KlMa (Theorem 12, Geometry of numbers)]

  • Solvable groups: [KlMa (Theorem 12, Methods from class field theory), FiKl]


Degree 8:

For imprimitive octic fields with a quartic subfield [CDO] compute huge tables using class field theory which cover all imprimitive groups and all possible signatures such that the corresponding field has a quartic subfield. These tables are not sufficient to find all minimal fields of that shape such that complex conjugation lies in a given class of involutions.

In [FiKl] the minima for octic fields having a quadratic subfield are computed. Altogether, we cover all possible signatures for all solvable groups.

  • Octic fields with a quartic subfield: [CDO]

  • Octic fields with a quadratic subfield: [FiKl]

  • S8 totally real: [KlMa]

  • 8T25, 8T36: [FiKl]


In the following table we list the non-solvable groups in degree 8. Here, we only know one minimum. The given discriminants are the smallest ones known to us.

Degree

Name

0

2

4

6

8

8T37

L(8)=PSL(2,7)

37822859361




235163942523136

8T43

L(8):2=PGL(2,7)

418195493

-1997331875



312349488740352

8T48

E(8):L7=AL(8)

32684089


293471161


81366421504

8T49

A(8)

20912329


144889369


46664208361

8T50

S(8)

1282789

-4296211

15908237

-65106259

483345053 (*)


(*) = proven minimum



Degree 9:

In degree 9 there are some partial results for imprimitive fields [DiOl]. Furthermore some totally real minimal discriminants are determined [Tak]. In a recent work, [JON] finished the computation of all minima for all solvable groups.


In the following table we list the non-solvable groups in degree 9.


Degree

Name

1

3

5

7

9

9T27

L(9)=PSL(2,8)

79082438656




16339134383250936197651486708561524809

9T32

L(9):3=P|L(2,8)

22663495936




342055275897107057019955703387435044864

9T33

A(9)

92371321


3200504329


11729467378561

9T34

S(9)

29510281

-109880167

453771377

-1904081383

9685993193


Degree 10:

For many imprimitive groups the minimal discriminants are determined in [DrJo]. In the meantime these computations have been extended to all fields with absolute discriminant smaller than 1.2*1011. Please have a look at Group10, select ?existence? and see the green arrows, which mark the already proven minima.



References:


  • [Bel] K. Belabas, A fast algorithm to compute cubic fields, Math. Comput. 66, No.219, 1213-1237 (1997).

  • [BFP] J. Buchmann, D. Ford, and M. Pohst, Enumeration of quartic fields of small discriminant, Math. Comput. 61, No.204, 873-879 (1993).

  • [BMO] A. Berge, J. Martinet, and M. Olivier, The computation of sextic fields with a quadratic subfield, Math. Comput. 54, No.190, 869-884 (1990).

  • [CDO] H. Cohen; F. Diaz y Diaz; M. Olivier, Tables of octic fields with a quartic subfield, Math. Comp. 68, No. 228, 1701-1716 (1999).

  • [Dia1] F. Diaz y Diaz, Discriminant minimal et petits discriminants des corps de nombres de degree 7 avec cinq places reelles, J. London Math. Soc., II. Ser. 38, 33-46 (1988).

  • [Dia2] F. Diaz y Diaz, Valeurs minima du discriminant pour certains types de corps de degree 7, Ann. Inst. Fourier 34, No.3, 29-38 (1984).

  • [DiOl] F. Diaz y Diaz, M. Olivier, Imprimitive ninth-degree number fields with small discriminants. With microfiche supplement. Math. Comp. 64, no. 209, 305-321 (1995).

  • [DrJo] E. Driver, J. Jones, Minimum discriminants of imprimitive decic fields. Experiment. Math. 19, No. 4, 475-479 (2010).

  • [FiKl] C. Fieker and J. Klüners, Minimal Discriminants for Fields with Frobenius Groups as Galois Groups, J.Numb.Th., 99, 318-337 (2003).

  • [FoPo1] D. Ford and M. Pohst, The totally real A5 extension of degree 6 with minimum discriminant, Exp. Math. 1, No.3, 231-235 (1992).

  • [FoPo2] D. Ford and M. Pohst, The totally real A6 extension of degree 6 with minimum discriminant, Exp. Math. 2, No.3, 231-232 (1993).

  • [FPD] D. Ford, M. Pohst, M. Daberkow, and H. Nasser, The S5 extensions of degree 6 with minimum discriminant, Exp. Math. 7, No.2, 121-124 (1998).

  • [IDPF] J. Carmelo Interlando, J. O. Dantas Lopes, T. Pires da Nobrega Neto, A.L. Flores; On the minimum absolute value of the discriminant of abelian fields of degree p2. J. Algebra Appl. 9, no. 5, 819-824 (2010).

  • [JON] J. Jones, Minimal solvable nonics, LMS J. Comput. Math., to appear.

  • [KlMa] J. Klüners and G. Malle, A Database for Field Extensions of the Rationals, LMS J. Comput. Math., 4, 182-196 (2001).

  • [Oli1] M. Olivier, The computation of sextic fields with a cubic subfield and no quadratic subfield, Math. Comput. 58, No.197, 419-432 (1992).

  • [Oli2] M. Olivier, Corps sextiques primitifs. IV, Semin. Theor. Nombres Bordx., Ser. II 3, No.2, 381-404 (1991).

  • [Poh1] M. Pohst, The minimum discriminant of seventh degree totally real algebraic number fields, Number Theory and Algebra; Collect. Pap. dedic. H. B. Mann, A. E. Ross, O. Taussky-Todd, 235-240 (1977).

  • [Poh2] M. Pohst, On the computation of number fields of small discriminants including the minimum discriminants of sixth degree fields, J. Number Theory 14, 99-117 (1982).

  • [SPD] A. Schwarz, M. Pohst, and F. Diaz y Diaz, A table of quintic number fields, Math. Comput. 63, No.207, 361-376 (1994).

  • [Tak] K. Takeuchi, Totally real algebraic number fields of degree 9 with small discriminant. Saitama Math. J. 17, 63-85 (1999).



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