# 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 C_{2}with the normal divisor C_{5}, - M.
_{5}the*metacyclic*Group of order 20, the semi-direct product of C_{4}with the normal divisor C_{5}, often also as Aff_{5}or Hol (C_{5}) 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 S_{5}-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 d**1991, Dept. of Comp. Sci., Univ. of Manitoba_{L.}< 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, - [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.

Back to Daniel C. Mayer's homepage.

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