On Totally Real Cubic Fields - American Mathematical Society · 2018-11-16 · On Totally Real Cubic Fields with Discriminant D < 107 By Pascual Llórente and Jordi Quer Abstract

MATHEMATICS OF COMPUTATIONVOLUME 50, NUMBER 182APRIL 1988, PAGES 581-594

On Totally Real Cubic Fields

with Discriminant D < 107

By Pascual Llórente and Jordi Quer

Abstract. The authors have constructed a table of the 592923 nonconjugate totally

real cubic number fields of discriminant D < 107, thereby extending the existing table

of fields with D < 5 x 105 constructed by Ennola and Turunen [4]. Each field is given by

its discriminant and the coefficients of a generating polynomial. The method used is an

improved version of the method developed in [8]. The article contains an exposition of

the modified method, statistics and examples. The decomposition of the rational primes

is studied and the relative frequency of each type of decomposition is compared with

the corresponding density given by Davenport and Heilbronn [2].

1. Introduction. A table of totally real cubic fields with discriminant D < D'

has been constructed by Godwin and Samet [5] for D' = 2 x 104. Angelí [1]

extended this table up to D' = 105 by using a similar method. These tables are not

complete (see [4] and [9]). A complete table for D' = 105 is constructed in [8] by a

different method. Finally, a third method is developed by Ennola and Turunen [4]

to compute a table with D' = 5 x 105. In this paper we shall describe an extended

table for D' = 107. The method used is an improved version of that developed in

[8]. Up to 5 x 105, this table agrees with Ennola and Turunen's.

The modified method and the new algorithm are described in Sections 2 and

3. Section 4 contains statistics and examples (Tables 1-6). The decomposition of

the rational primes is studied using the congruential criteria given in [6] and [7].

In Section 5 we compare the relative frequency of each type of decomposition for

different primes with the corresponding density given by Davenport and Heilbronn

in [2] (Tables 7-11).

Computations were done on an IBM 3360 owned by the Universität de Barcelona

and a VAX-8600 owned by the Facultat d'Informàtica de Barcelona.

2. The Improved Method. The method used to compute our table of totally

real noncyclic cubic fields is similar to the one described in [8] but improved in

several ways. In this section we recall the main ideas in [8] and we explain the

modifications introduced. For every prime p G Z and integer m, vp(m) denotes the

greatest integer k such that pk divides m.

Each triple of conjugate noncyclic fields is defined by a polynomial of the type

(1) f(a,b,X) = X3 -aX + b,

Received September 22, 1986; revised February 17, 1987 and August 7, 1987.

1980 Mathematics Subject Classification (1985 Revision). Primary 11R16, 11-04.

Key words and phrases. Real cubic fields.

©1988 American Mathematical Society

0025-5718/88 $1 00 + $.25 per page

581

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use

582 PASCUAL LLÓRENTE AND JORDI QUER

where a and b are positive integers such that

(2) f(a, b, X) is irreducible in Q[X],

(3) there is not a prime p with vv(a) > 2 and vp(b) > 3.

Then, each field on the triple is isomorphic to the cubic field K = K(a,b) = Q(6),

where 9 is a root of the polynomial f(a, b, X). The descriminant of f(a, b, X) is

(4) D(a, b) = 4o3 - 2762 = DS2,

where D — D(K) is the descriminant of the cubic field K and S > 0 is the index

of 9. It is known (cf. [3] or [7]) that

(5) D = D(K)=dT2,

where d is the discriminant of Q(VD) and T = 3mT0 with 0 < rn < 2 and T0 > 0

is a square-free integer having no common factor with 3d. Note that d > 1, since

we are assuming that K is totally real and noncyclic.

Consider the congruences

(6) a = 3 (mod 9), b = ±(a -1) (mod 27).

We have the following result (cf. [3, p.112]).

THEOREM 1 (VORONOI). (i) If the congruences (6) are not satisfied, then S

is the greatest positive integer whose square divides D(a, b) for which there exist

integers t, u and v such that

(7) -S/2<t<S/2, Zt2~a = uS, t3 -at + b = vS2,

and 1,9, ß, with ß = (92 +t9 + t2 — a)/S, is a basis for the integers of K.

(ii) // the congruences (6) are satisfied, then S = 27S', and S' is the greatest

positive integer whose square divides D(a,b)/729 for which there exist integers t,u

and v such that

(8) -3S'/2<t<3S'/2, 3t2-a = 9uS', t3 - at + b = 27vS'2,

and 1, ip, ß, with tp — (9 - £)/3 and ß = (92 +t9 + t2 - a)/9S, is a basis for the

integers of K.

It follows that by choosing a minimal t in (7) or (8) there is a unique quadruple

(S,t,u,v) of integers associated with each pair (a, b). The main idea in [8] is to

associate a positive definite binary quadratic form F(a,b) with each pair (a, b),

whose coefficients are given in terms of a, S, t, u and v. We shall give an alternative

definition for the quadratic form associated with the pair (a, b) (see D, below). We

repeat here the definition and some of the properties of F(a, b) given in [8], to make

our exposition more self-contained.

If the congruences (6) are not satisfied, then Tr(l) — 3, Tr(0) = 0 and Tr(/3) = u

are the traces of the integers in a basis for the integers of K. So the integers 7 € K

with zero trace are given by

(9) 7 = (Zx92 + Z(xt + Sy)9 - 2ax)/SS; x,yeZ, 3 | ax.

Let 7 t¿ 0 be such an integer. Its minimum polynomial is f(a', —N(~i),X), N(-y)

being the norm of 7 and a' = (3u2 - 27tv)x2/9 + (Zut - 9vS)xy/3 + ay2.


ON TOTALLY REAL CUBIC FIELDS WITH DISCRIMINANT D < 107 583

If 3 f u, since Tr(9) = 0, we have 3 | x. Let z = x/3; then

a' = (3u2 - 27tv)z2 + (3ut - 9vS)zy + ay2.

If 3 | u, let u = 3w; then

a' = (3w2 - 27tv)x2 + (ut - 3vS)xy + ay2.

If the congruences (6) are satisfied, then

a = 3oi with ai = 3a2 + 1 and a2 €E Z,

ct = 3ui+n with |r/| < 2 and ui,n G Z,

t = 3ii + 6 with 6 = ±1 and t\ € Z.

The integers 7 € Ä" with zero trace are given by

(10) 7 = (x92 + (tx + 3fiSx + 9Sy)9 - 2axx)/9S; x,y&Z, p = ne.

Let 7 t¿ 0 be such an integer. Its minimum polynomial is f(a', -N(^),X), N(i)

being the norm of 7 and

a' = (uui + 2nui —vt + puti - 3pvS' + u2a2 + p2)x2 + (ut + 2pai - 9vS')xy + ay2.

With these notations, the quadratic form F(a, b) associated with the pair (a, b)

is defined in [8] as follows:

DEFINITION 1. (i) If the congruences (6) are not satisfied and 3 \ u, we define

F(a, b) = (3u2 - 27tv, 3ut - 9vS, a).

(ii) If the congruences (6) are not satisfied and u = 3w, we define

F(a,b) = (3w2 - 3tv,ut - 3vS,a).

(iii) If the congruences (6) are satisfied we define

F(a, b) = (mti + 2nui — vt + puti — 3uvS + u2a2 + p2,ut + 2pai — 9vS, a).

Then F(a, b) represents precisely those integers a' > 0 for which there exists an

integer b' such that K(a',b') is isomorphic to K(a,b). Then, if K(a,b) «a K(a',b'),

the corresponding associated forms F(a, b) and F(a', b') represent the same integers

and, consequently, they are equivalent (cf. [8] or [10]).

THEOREM 2. Let F — F(a, b) be the form associated with the field K = K(a, b)

of discriminant D — D(K). Then the discriminant D(F) of the form F is

D(F) = -D/3 if 27 \D,

D(F) = -3D if 27 f D.

Proof. One computes D(F) from Definition 1. Then D(F) = -D/3 in case (ii),

and D(F) = —3D in cases (i) and (iii).

In case (ii) of Definition 1, we have a = 3ai (since u = 3w) and then 27 | DS2

in (4). If 3 f S then 27 | D; else, easy congruential considerations show that if

3 I u and 9 | S, then the congruences (6) are satisfied, hence we must have S = 3Si

with 3 f Si and D = 3Di. From (7) it follows that 01 = t2 ( mod 3) and b = 2t3

(mod 3). Then DiSf = 4a? - b2 = 0 ( mod 3), so 9 | D and from Theorem 2 of [7],

27 must divide D.



Conversely, from Theorem 2 of [7] and easy congruential considerations one can

show that if 27 | D then case (ii) of Definition 1 holds. D

From this result and the elementary theory of reduced positive definite quadratic

forms we obtain

THEOREM 3. Let K be a totally real noncyclic cubic field of discriminant D =

D(K). Then K «s K(a,b) for some integer a with

a < y/D/3 if 27 | D,

a<\ÍD if 27 \ D.

In this case, we have

S <S(a) = 2s/a~¡3 if 27 \D,

S<S(o) = 2v/ö" if 27 i D.

As a consequence, a table of all totally real noncyclic cubic fields of discriminant

D = D(K) < D' can be constructed from a finite number of pairs (a,b), carrying

out the following steps:

- Elimination of all pairs not satisfying (2) or (3).

- Decomposition of D(a, b) as in (4) for the remaining pairs.

- Elimination of all pairs not satisfying the bounds of Theorem 3.

- Elimination of isomorphic fields.

In this way, a table with D' = 105 was constructed (see [8] and [9]); but this

algorithm is too inefficient (about 70 hours of computer time were needed to con-

struct that table) to be applied to much greater D'. Moreover, the computation of

D(a, b) requires multiple-precision arithmetic. We explain below the improvements

introduced in the method. With them, the method becomes much more efficient.

A. Irreducibility of f(a, b, X). It is convenient to observe that each of the follow-

ing conditions implies the irreducibility of f(a,b,X) in Q[X]:

(11) 1 < vp(b) < Vp(a) for some prime p,

(12) a = b = 1 ( mod 2),

(13) a = 1 ( mod 3) and 3 + 6.

In [8] one uses the fact that f(a,b,X) is reducible if and only if it has a root

m G Z. In this case, m | b, and an elementary study of f(a,b,X) shows that such

a root m must satisfy

0 < m < y/a~ or \/E < —m < \/ä + \Ja/3.

B. Computation of D(K) and S. In [8], D = D(K) and S were computed from

D(a, b) using Theorem 1. This was certainly the most laborious part of the method.

The results in [7] simplify this computation. Indeed, if (11) is satisfied for a prime

p, then p | T and (11) is also a necessary condition for p | T if p ^ 3; the factor

3m of T can be determined from certain congruential conditions involving a and b

(see [7, Theorems 1 and 2]). In this way, T is computed and, eventually, a divisor

So of S is also obtained. It is also easy to compute v2(D) and v2(S) from a and

b in case 2 \T. Now, every new integer m whose square divides D(a,b) will be

a new factor of S, i.e., m | S if and only if D(a,b) = 0 ( mod m2). Also, these



computations sometimes give several prime factors of d; in fact, p \ d if any of the

following conditions is satisfied:

(14) 1 = vp(a) < vp(b),

(15) vp(D(a,b)) is odd.

These conditions are also necessary for p \ d, except if p = 3. In this case, some

additional congruential conditions must be considered (see [7, Theorem 1]). Using

the bound S < S(a) in Theorem 3, one can easily compute S (and then D) or

eliminate the pair (a,b).

C. Eliminating Superfluous Fields. With the help of Theorem 3 we can eliminate

all pairs (a, b) for which S > S(a). We can eliminate also the pairs with D < D(a),

where

(16) D(a)=9a2 or D(a) = a2, according as 27 | D or not,

because in this case K(a,b) ss K(a',b') for some a' < a. The determination of the

bounds S(a) and D(a) (i.e., if 27 | D or not) follows from the computations in B.

In many cases, this elimination can be done by knowing only some factors of D and

S, and it is not necessary to compute S and D completely as in [8].

D. Eliminating Isomorphic Fields. If we obtained two fields Kx = K(ai,bi) and

K2 = K(a2,b2) with the same discriminant D, we have to test if Kx sa K2 or

not. As in [8], this can be done by studying their associated quadratic forms. The

method has been improved at this point too. We start giving a new definition of

the associated quadratic form:

DEFINITION 2. (i) In the first case of Voronoi's Theorem, let B - (3b-2at)/S.

Then we define

F* = F*(a,b) = (a,B,C) if 27 | D,

F* = F*(a,b) = (a,3B,C) if 27 \ D,

where C is determined by the condition D(F*) = -D/3 or D(F*) — -3D, accord-

ing as 27 | D or not.

(ii) In the second case of Voronoi's Theorem, let a' — a/3, let w be the only integer

with |u| < 2 and « = t(t2 - a')/3S' ( mod 3) and let B = -2pa' + (b - 2a't)/3S'.

Then we define

F* = F*(a,b) = (a,B,C),

where C is determined by the condition D(F*) = -3D.

It is immediate to see that the associated form F*(a,b) given in this definition

is equivalent to the associated form F(a,b) given in Definition 1. An advantage

of this new definition is that it is not necessary to compute u and v in Voronoi's

Theorem. Moreover, if a is minimum, the reduced equivalent quadratic form is

easily obtained from a and B; it suffices to find n G Z such that

(17) B' = B + 2an and \B'\<a.

Then the reduced form will be F' — (a,B',C), where C" is determined by the

discriminant.

The following theorem permits us to eliminate a large number of isomorphic

fields.



THEOREM 4. Let Ki = K(ai,bi) satisfy the conditions of Theorem 3, and let

F' = (ai,B',C) be the reduced form equivalent to its associated form. Then there

exists a pair (a2,b2) with a2 > ai, and b2 ^ bi in case a2 = ai, such that the field

K2 = K(a2,b2) satisfies the conditions of Theorem 3 and is isomorphic to Ki if

and only if D(C) < D(Ki). In this case, we must have C = a2, and there is only

one such pair.

Proof. We know that a2 = F'(x,y) must be an integer represented by F', and

by (3) we have that y ^ 0. It is easy to see that C" = .F(0,1) is the only integer

a represented in that way by F' for which the condition D(a) < D(Ki) can be

satisfied. D

Theorem 4 is not sufficient to eliminate isomorphic fields if there exists a third

field A3 = K(ü3,bs) satisfying the conditions of Theorem 3, with the same dis-

criminant as Ki and K2 and with a2 = a^ = C (D = 77844 is an instance of this

case). Then we have to decide whether Ki « K2 or Ki sa ä"3. For this we can

proceed as in [8]: The representation of a2 by F(ai,bi) determines an integer 7 in

Ki with zero trace, whose minimal polynomial is f(a2, —N(^),X). Then Ki k, K2

or Ki sa A3 according as |iV('y)] is equal to b2 or 63.

Using the definition of 7 (see (9) and (10)) and the transformation taking

F(ai,6i) into its reduced associated form F' = (a1,5',a2), and computing ex-

plicitly the norm N(*i) of the integer 7, we obtain

THEOREM 5. Let Ki = K(ai,bi) satisfy the conditions in Theorem 3 with

D(Ki) = D. Let F' = (ai,B',a2) be the reduced form equivalent to its associated

form F — F(ai,bi) with D(a2) < D, and let 7 be the null trace integer in Kx

determined by the representation of a2 by F. Then:

(i) IfD = 27D' and ax = 3a, then

N{i) = ((2aa2 - D')S2 - 4a3 +bx(t- nS)(ai - (t - nS)2))/S3.

(ii) If 27 ^ D and the congruences (6) are not satisfied, then

N{i) = ((2axa2 - D)S2 - 4a? + 61 (3* - r»S)(9ai - (3i - nS)2))/S3.

(iii) If ai = 3a and the congruences (6) are satisfied, then

N(1) = ((2a1a2-D)27S"2-4a3+bi(t+3uS'-9nS')(ai-(t+3uS'-9nS')2))/(9S')3.

Here, S, S', t and u are the integers used in Definition 2 and n is the integer deter-

mined by (17).

3. The New Algorithm. The method described in the previous section pro-

vides an easy computer-programmable algorithm to construct a table of the totally

real cubic fields K with discriminant D = D(K) < D' for a given D'. We shall

now describe the algorithm used by the authors.

We take all integers a with 4 < a < y/TP and, for each, we take the integers b

with 1 < b < 2(a/3)3//2. For each pair (a, b) we proceed as follows:

Step 1. Compute M = g.c.d.(a,6) and work with every prime factor p of M.

During these computations:

(a) The pairs not satisfying (3) are eliminated.



(b) T0 is always obtained and T is also obtained if 3 | M (as is explained in B of

Section 2). In case 3 | M, the bounds S (a) and D(a) are determined.

(c) vp(D) and vp(S) are computed using (b) and (14).

(d) In some cases, irreducibility of f(a,b,X) is proved by (11).

Step 2. If irreducibility of f(a,b,X) has not been proven in Step 1(d), use the

other results in A of Section 2 to eliminate the pairs with /(a, b, X) reducible.

Step 3. If 3 -f M, compute v$(D) and t>3(S), using the congruential conditions

given in [7]. So, T is obtained and the bounds S(a) and D(a) are determined.

(Remark. During the computations, divisors So of S and Do of D are obtained.

If So > S(a) or Do > D', the pair (a,b) is eliminated.)

Step 4. If 2 f M, compute v2(D) and v2(S), using the congruential criterion

given in [7].

Step 5. Let Si = S (a)/So- For every prime p with p \ 6M and p < Si, examine

D(a,b) modulo p2 (if p2 < Si then examine D(a,b) modulo p4). In this way:

(a) Vp(D) and vp(S) are computed using (15).

(b) Eventually, Do, So and Si are modified.

(c) The final S0 is the greatest S in (4) with S < S (a).

Step 6. Let Di = D(a,b)/S(2. If Dx > D', the pair (a, b) is eliminated. Indeed,

in this case we have D > D' or S > S (a).

Step 7. Let D2 = Di/D0T2. The pair (a, b) is eliminated in the following cases:

(a) If D2 is not square-free (in this case, S > S(a)).

(b) If D2 = D0 = 1 (in this case, d = 1).

(c) IfDi <D(a).

Step 8. If the pair (a, b) has not been eliminated in the preceding steps, K =

K(a,b) is a cubic field with discriminant D = Di < D' satisfying the conditions of

Theorem 3. During the process, d — D0D2, T and S = Sr, have been computed.

Record these data in a file.

Step 9. Data in that file are ordered with increasing discriminant D without

altering the order of their generation among the fields with the same discriminant.

Step 10. Eliminate isomorphic fields in the file by using the results of D in

Section 2. To do this, proceed as follows: Let Ki = K(ai,bt), i = 1,...,N,

be all the fields in the file with the same discriminant D. According to Step 9,

we have ai < a2 < • ■ • < a^. Compute the reduced quadratic form (ai,i?,C)

equivalent to the quadratic form associated with the pair (ai,6i). By Theorem 4,

Ki is isomorphic to some Ki with i > 1 (and only to one of them) if and only if

D(C) < D, and in this case, C = a,. If C = ai for only one t > 1, then eliminate

the field Ki. If C = a¿ for more than one i > 1, then compute N(i) by using

Theorem 5 and eliminate the field Ki = A"(a¿,6¿) for which í>¿ = |JV(7)|. Proceed

in the same way with the next noneliminated field until the set of all fields with

discriminant D is completely purged of isomorphic pairs.

This algorithm is very efficient. A table for D' = 105 can be constructed in less

than a minute of computer time, and with D' = 5 x 105 in about five minutes.

We have used it with D' = 107, and the total computer time required was about 5

hours. Almost all time was spent in the generation of the fields K(a, b) (Steps 1 to

8). The purge of isomorphic fields in Step 10 eliminated around 10% of the stored

fields and required about 8 minutes.



4. Totally Real Cubic Fields with Discriminant D < 107. The table con-

taining the 592422 nonconjugate totally real noncyclic cubic fields with discriminant

less than 107 consists of 10 sectors, the kth containing the fields with discriminants

between I06(k — I) and 106fc. For each field K = Q(9), its discriminant D, the

coefficients a and 6 of a polynomial Irr(#, Q) = f(a, b, X) and the integers 5 and T

are listed. Other data obtained during the computations were not stored for lack

of space. Separately, we have constructed a table of the 501 cyclic cubic fields with

discriminant less than 107.

TABLE 1

Number of cubic fields with discriminant 0 < D < 107

Sector

1'2

34

5IS

7

8

910

Noncyclic

54441

577775878759266597385999460376605076070560831

Cyclic

159

6747

44

36

3627

352525

Total

54600

57844588345931059774

6003060403

605426073060856

Noncyclic

(accum.)

54441112218171005230271290009350003410379470886531591592422

Cyclic

(accum.)

159226273317353389416451476501

Total

(accum.)

54600112444171278230588290362350392410795471337532067592923

TABLE 2

Number of discriminants 0 < D < 107 with 2 associated nonconjugate cubic fields

Sector

1

234

56789

10

Noncyclic

166

223206218231221

241224

262239

Cyclic

37

18

10

11

999

69

Total

203

241

216

229240230250230271247

Noncyclic

(accum.)

166

389595813

104412651506173019922231

Cyclic

(accum.!

37

556576

8594

103109

118

126

Total

(accum.;

203

444

660889

1129

1359160918392110

2357

Table 1 gives the number of cubic fields K by sectors. We give for each sec-

tor k the number of noncyclic cubic fields, cyclic cubic fields and cubic fields with

discriminant 106(fc - 1) < D < 106/c and with discriminant 1 < D < 106fc (accu-

mulated). Tables 2, 3 and 4 are similar for the number of discriminants with N

associated nonconjugate fields, for N — 2, 3 and 4, respectively.

There are five discriminants D < 107 with N associated nonconjugate fields

for N > 4. And we have N = 6 for all of them. In Table 5 we have listed

these discriminants, their decomposition D — d32mTr2 and the coefficients a and

b of the polynomials f(a,b,X) defining the six corresponding fields. Among the

discriminants corresponding to noncyclic fields in Table 4, there are nine with T > 1;

these discriminants are listed in Table 6 in a way similar to those in Table 5.



TABLE 3


Sector

1

2

34

56

789

10

Noncyclic

343468557552

568601

624622582626

Noncyclic

(accum.)

343811

13681920248830893713433549175543

TABLE 4


Sector

1

2

34

56789

10

Noncyclic

161

233

255

284283

293

311

348364347

Cyclic Total

162234

256285284

294

311350364347

Noncyclic

(accum.)

161394649933

121615091820

216825322879

Cyclic(accum.

Total(accum.)

162

396652937

1221

151518262176

25402887

From Tables 4 and 6 we observe that there are 2879 discriminants D = d,

1 < D < 107, with four associated noncyclic fields. From class field theory we can

conclude that there are 2879 real quadratic fields with discriminant d < 107 whose

ideal class group H(d) has 3-rank equal to 2. A complete study of the H(d) having

3-rank > 1 for d < 107 will be given in a later paper.

5. On Davenport and Heilbronn's Densities. The total number of noncon-

jugate cubic fields with discriminant less than 107 is 592923, giving the empirical

density 0.05929. Table 1 easily permits the computation of this empirical density

in each sector. Davenport and Heilbronn prove in [2] that the asymptotic value

is (12c(3))_1 « 0.06933. Thus the convergence is very slow, as was noted in [11].

In [2], Davenport and Heilbronn obtain also the asymptotic value of the density

of each type of decomposition of a rational prime p in cubic fields. The values

obtained for them are:

l/w for p = P3,

p/w for p = PQ2,

p2/3w for p = P,

p2/2w for p = PQ,

p2/6w for p = PQR,

with w = p2 +p+ I.



Table 5

Discriminants 0 < D < 107 with 6 associated cubic fields

D=32"T d

Coefficients of the polynomial f(a,b,X~)

3054132 84837

96102150

210336366

134210

622112212442674

4735476 9 2 131541

108114168252288648

106210726

129218344772

5807700 81 10 717

180270360450540720

60990

246028504740

6690

6367572 81 2 19653

126

342396450540720

246174

2994364238523012

9796788 81 2 30237

144216540648756918

282204

320452207794

5742

Using the results of [6] and [7], we have computed the decomposition type of the

rational primes p for 2 < p < 181 in each of the 592422 noncyclic cubic fields in

the table. In particular, there are 46532 noncyclic cubic fields K with D(K) < 107

having 2 as its common index divisor, i.e., with the rational prime 2 decomposing

completely. We give in Tables 7, 8, 9 and 10 the empirical density of each type of

decomposition by sector and its asymptotic value for p = 2,3,5 and 7, respectively.

In Table 11 we give the empirical density and its asymptotic value of each type of

decomposition for the primes p with 2 < p < 100 in all fields in the table.



TABLE 6

Discriminants 0 < D < 107 with 4 associated cubic fields and T = 32mTo > 1

D=3*"Td 32"' T,

Coefficients of the polynomial f(a,b,X)

1725300 51 10 21390

180270360

210780

15302580

2238516 81 14 141126252252378

462546

14282814

2891700 il 10 35790

180180360

30660870

1290

4641300 81 10 573180270540540

420117015904140

6810804 51 14 429126378378504

210130223944326

7557300 81 10 933270450720810

6302550

6608730

7953876 81 14 501126252378756

421092

7987308

8250228 9 14 4677336630756840

1694266

79248764

8723700 51 10 1077270450540900

90363034208700



TABLE 7

Type of decomposition of the prime 2 in the

noncyclic cubic fields with discriminant 0 < D < 107 (%)

Sector PJ PQ2 PQ PQR

1

234

56

789

10

15.79715.37815.17315.16715.21115.04815.11515.03814.90015.163

27.54027.80027.99627.82428.00428.01428.0282802328.15327.951

21.45820.86020.61220.59020.45920.46720.41220.37920.33120.210

28.33528.38528.51028.50428.41228.44328.42928.47128.44628.532

6.8707.5777.7097.9157.9138.0278.0168.0888.1718.144

All Table 15.192 27.938 20.567 28.448 7.855

Theoretical 14.286 28.571 19.048 28.571 9.524

TABLE 8


noncyclic cubic fields with discriminant 0 < D < 107(%j

Sector PQ2 PQ PQR

1

2

34

567

89

10

8.4168.154

8.1628.0608.1518.0828.1098.0557.881

8.162

22.25722.447

22.49322.54622.66122.61222.65822.53822.652

22.628

25.98825.27125.03824.94324.84224.74624.71024.66324.65924.458

34.71834.69434,68334.68334.58434.71034.46934.76834.816

34.670

8.6209.4359.6259.7689.7639 849

10.054

9.9769.993

10.082

All Table 8.120 22.553 24.919 34.679 9.729

Theoretical 7.692 23.077 23.077 34,615 11.538

TABLE 9



Sector PQ2 HQ PQR

1

2

34

56

789

10

3.4773.4153.3753.3903.2963.4343.3793.34233113.362

15.37315.65015.70215.66515.79715.75215.79315.78515.67315.869

30.14829 23029.10328.82128.85128.67628.72828.64128.58728.558

40.55840.53340.47840.48740.55240.38740.40240.38440.552

40.307

10.444

11.17211.34111.63711.50411.75111.69811.84811.877

11.905

All Table 3.377 15.709 28.920 40.463 11.531

Theoretical 3.226 16.129 26.882 40.323 13.441



TABLE 10



Sector PQ2 PQ PQR

1

2

3

45

6

78

910

1.8591.784

1.8291.7941.8631.7841.8471.7781.8171.863

11.71211.87311.95511.86311.86812.01011.95312.05111.93111.843

31.99230.96730.86930.67530.58730.65630.55530.13430.50330.513

43.14843.34142.99243.36643.05342.98443.03543.39343.07142.942

11.28912.03412.35512.30212.62812.56612.60912.64312.67812.840

All Table 1.822 11.908 30.731 43.131 12.408

Theoretical 1.754 12.281 28.655 42.982 14.327

TABLE 11

Type of decomposition of the rational primes p < 100 in the


Prime

EmpiricalTheore-

tical

PQ2

EmpiricalTheore-

ticalEmpirical

Theore-

tical

PQ

EmpiricalTheore-

tical

PQR

EmpiricalTheore-

tical

2

3

57

1113

17

19

23

2931

37

41

43

47

535961

67

71

7.3

79

83

8997

15.192

8.1203.3771.822

0.7760.5590.3320.2650.1880.120

0.102

0.0690.0630.051

0.0430.0370.0280.025

0.0210.021

0.0180015

0.0150.0130.013

14.2867692

3.2261.754

0.7520.5460.326

0.2620.1810.1150.101

0.0710.0580.0530.0440.0350.028

0.0260.0220.0200.0190.0160.014

0.0120.011

27.93822.55315.70911.9088.0156.8935.3594.8224.0173.216

2.9962.5362.3102.209

2.0081.7981.627

1.5801.4331.3351 3081.1961 1481.080

0.989

28.57123.07716.12912.281

8.2717.1045.5374.9874.159

3.3303.122

2.6302.3802.2722.0821.8511.6661.6121.4701.3891.3511.2501.190

1.1111.020

20.56724.91928.92030.731

32.30032.7053321833.39133.6143381133.89334.01734.02234.057

34.119

34.10434.229

34.16134.21134.10834.18134.22734.21234.20234.257

19.04823.07726.88228,65530.32630.78331.37931.58431 88732.18532.25932.43332.52132.55932.62432.70532.76932.78732.83632.86432.877329113293232.95932.990

28.44834.67940.46343.13145.62846.28647.158

47.47347.90048.34048.42948.68648.82448.863

48.92649.0784906549.15849.20349.40349.31349.33849390

49.41649.400

28.57134.61540.32342.98245.48946.17547.06847.37547.83048.2784838948.65048.78148.83848.93749.05749.153

49.18149.25449.29649.31549.367

49.39849.43849.485

7.8559.729

11.53112.40813.28013.55813.93314.04914.28114.51214.58014.69314.78314.81914.904

14.98415.05115.07715.13215.13215.17915.224

15.23415.28915.340

9.52411.53813.44114.32715.16315.39215.68915.79215.94316.09316.13016.21716.26016.279

16.31216.35216.38416.39416.41816.43216.43816.45616.46616.47916.495



Departamento de Matemática Aplicada

Edificio Matemáticas, Planta 1

Universitad de Zaragoza

50009 Zaragoza, Spain

Departament de Matemàtiques

Facultat d'Informàtica de Barcelona

Universität Politécnica de Catalunya

Pau Gargallo,5

08028 Barcelona, Spain

1. I. O. ANGELL, "A table of totally real cubic fields," Math. Comp., v. 30, 1976, pp. 184-187.2. H. DAVENPORT & H. HEILBRONN, "On the density of discriminants of cubic fields II,"

Proc. Roy. Soc. London Ser. A, v. 322, 1971, pp. 405-420.3. B. N. DELONE & D. K. FADDEEV, The Theory of Irrationalities of the Third Degree, Trudy

Mat. Inst. Steklov., vol. 11, 1940; English transi., Transi. Math. Monographs, vol. 10, Amer. Math.

Soc, Providence, R.I., Second Printing 1978.

4. V. ENNOLA k R. TURUNEN, "On totally real cubic fields," Math. Comp., v. 44, 1985,

pp. 495-518.5. H. J. GOODWIN & P. A. SAMET, "A table of real cubic fields," J. London Math. Soc, v. 34,

1959, pp. 108-110.6. P. LLÓRENTE, "Cubic irreducible polynomials in ZP[XJ and the decomposition of rational

primes in a cubic field," Publ. Sec. Mat. Univ. Autónoma Barcelona, v. 27, n. 3, 1983, pp. 5-17.

7. P. LLÓRENTE & E. NART, "Effective determination of the decomposition of the rational

primes in a cubic field," Proc. Amer. Math. Soc, v. 87, 1983, pp. 579-585.

8. P. LLÓRENTE & A. V. ONETO, "Cuerpos cúbicos," Cursos, Seminarios y Tesis del PEAM,

n. 5, Univ. Zulia, Maracaibo, Venezuela, 1980.

9. P. LLÓRENTE & A. V. ONETO, "On the real cubic fields," Math. Comp., v. 39, 1982,

pp. 689-692.10. M. SCHERING, "Théorèmes relatifs aux formes binaires quadratiques qui représentent les

mêmes nombres," Jour, de Math. (2), v. 4, 1859, pp. 253-270.

11. D. SHANKS, Review of I. O. Angell, "A table of totally real cubic fields," Math. Comp., v.

30, 1976, pp. 670-673.


Documents

On Totally Real Cubic Fields - American Mathematical Society · 2018-11-16 · On Totally Real Cubic Fields with Discriminant D < 107 By Pascual Llórente and Jordi Quer Abstract