Why is 1953 125 not a square number?

and the Galois groups of their normal bodies


Designations:

Algebraic number fields of 5. Absolute degrees above the field Q of the rational numbers are denoted as. You enter 3 possible Signatures (r, s) where r denotes the number of real embeddings and s the number of pairs of conjugate complex embeddings: as the body of the signature (5,0) or the body of the signature (3,1) or the body of the signature (1,2).


Steps of the systematic investigation:

  • 1955, H.Cohn: complete lists of small quintic discriminants for all signatures and generating polynomials.
  • 1957, J. Hunter: minimal quintic discriminants for each signature: +14641, -4511, +1649; Expansion of Cohn's lists.
  • 1974, P. Cartier and Y. Roy: Solids in Cohn's lists with matching discriminants are isomorphic.
  • 1982, D. G. Rish: complete tables from easy and triple real quintic solids.
  • 1984, K. Takeuchi: all total real field 5th degree with discriminants d < 150000="" (insgesamt="" 21="" körper).="">
  • 1991, F. Diaz y Diaz: [1] table of 1077 total real quintic field with a discriminant d < 2000000;="" untersuchung="" der="" 5="" möglichen="" galois-gruppen="" des="" normalkörpers.="">
    Takeuchi discovered 2 errors: There are no quintic fields with discriminants d = 107653 or 146205, because the polynomials given are reducible over Q.
  • 1994, A. Schwarz, M. Pohst and F. Diaz y Diaz: [2] Tables of quintic bodies with discriminants d < 20000000="" für="" signatur="" (5,0),="" d=""> -5000000 for signature (3.1) and d < 5000000="" für="" signatur="" (1,2);="" verteilung="" der="" 5="" möglichen="" automorphismen-gruppen="" des="" normalkörpers.="">
  • 1998, M. Pohst and K. Wildanger: [3] Basic units, regulators, order and structure of the ideal class groups of all fields in the table from 1994; Statistics of the occurring class group structures.

Galois groups and statistics:

The following table shows the Population of the 5 possible Galois groups of the normal body of totally real quintic bodies:

  • C.5 the cyclical Group of order 5,
  • D.5 the The the-Group of order 10, the semi-direct product of C2 with the normal divisor C5,
  • M.5 the metacyclic Group of order 20, the semi-direct product of C4 with the normal divisor C5, often also as Aff5 or Hol (C5) designated,
  • A.5 the alternating Group of order 60,
  • S.5 the symmetrical Group of order 120.

According to N. H. ABEL's theorem, the last two are groups not resolvable. The primitive element of the corresponding quintic bodies cannot therefore be represented by radicals via Q.

The cyclical and dihedral cases have been known from my table [4] since 1991:

The 5 cyclic quintic solids have the guides f = 11, 25, 31, 41 and 61. The next ones would be 71, 101, 131, et c.

For the 26 quintic fields with a dihedral normal field N of degree 10, the cyclic quintic relative expansion is N | k from N over its square subfield k without exception unbranched with leader f = 1. The assigned quadratic discriminants d (k) extend from 401, 817, et c. to 4441, 4444. (These are the minimal quadratic discriminants with 5-class rank r = 1.) The next would be 4504, 4757, 4865, et c. The smallest branched cases are already above the limit 20,000,000: they have the leader f = 5 and assigned quadratic discriminants d (k) = 185, 280, 460, 520, etc. (congruent 0 modulo 5).

It is interesting to compare the results for totally real quintic discriminants with the results from [3] for the significantly more dense quintic discriminants of mixed signatures:

The non-resolvable S5-Structures are extremely dominant in all signatures of quintic number fields.

Among the 81 metacyclic According to my theory, bodies of the signature (1,2) come in [5] only 8 pure quintic bodies before:

  • d = 50000 (radicand R = 2),
  • d = 162000 (R = 2 * 9),
  • d = 253125 (R = 3),
  • d = 300125 (R = 7),
  • d = 1953125 (R = 5),
  • and a triplet of non-isomorphic quintic fields with a matching discriminant d = 4050000 for R = 2 * 3, 4 * 3, 16 * 3.

Literature:

  • [1] F. Diaz y Diaz, A table of totally real quintic number fields, Math. Comp. 56 (1991), 801-808 and Supplements section S1-S12
  • [2] A. Schwarz, M. Pohst and F. Diaz y Diaz, A table of quintic number fields, Math. Comp. 63 (1994), 361-376
  • [3] M. Pohst and K. Wildanger, Tables of unit groups and class groups of quintic fields and a regulator bound, Math. Comp. 67 (1998), 361-367
  • [4] D. C. Mayer, List of discriminants dL. < 4="" *="">10 of totally real quintic fields L, cyclic or with dihedral normal closure N of degree 10, arranged according to their multiplicities m and conductors f, 1991, Dept. of Comp. Sci., Univ. of Manitoba
  • [5] D. C. Mayer, Discriminants of metacyclic fields, Canad. Math. Bull. 36(1) (1993), 103-107

The calculations in [3] would be without the KANT package from POHST A.M.T. been unthinkable.


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