Computing all monogeneous mixed dihedral quartic extensions of a quadratic field

I

STVÁN

G

AÁL

G

ÁBOR

N

YUL

Computing all monogeneous mixed dihedral quartic

extensions of a quadratic field

Journal de Théorie des Nombres de Bordeaux, tome

13, n

o

_{1 (2001),}

p. 137-142

<http://www.numdam.org/item?id=JTNB_2001__13_1_137_0>

© Université Bordeaux 1, 2001, tous droits réservés.

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Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques

Computing

all

_monogeneous

mixed

dihedral

quartic

extensions of

a

quadratic

field

par ISTVEÁN

GAÁL*

et

GÁBOR

NYUL

RÉSUMÉ. Soit M un _corps

_quadratique

réel. Nous donnons un

algorithme rapide

pour déterminer tous les corps quartiques dié-draux K avec

_signature

_mixte,

monogènes (i.e.

_ayant des bases

d’entiers {1,03B1,03B12,03B13})

et contenant M comme _sous-corps. Nous

déterminons

_également

tous les

_{générateurs}

03B1 des bases dans K

ayant cette forme. Notre

_algorithme

combine un résultat récent

de Kable

_[9]

avec

l’algorithme

de

Gaál,

de Pethö et de Pohst

_[6],

[7].

On

_applique

la méthode à M =

Q(~2), Q(~3), Q(~5).

ABSTRACT. Let M be a _given real

_quadratic

field. We _give a fast

_algorithm

for

_determining

all dihedral _quartic fields K

with mixed _signature

_having

_power

_integral

bases and

contain-ing M as a subfield. We also determine all _generators of _power

integral

bases in K. Our

_algorithm

combines a recent result

of Kable

_[9]

with the

_algorithm

of

_Gaál,

Pethö and Pohst

_[6],

[7].

To illustrate the method we

performed

_computations for

M =

_{Q (~2),}

_{Q (~3),}

_Q(~5).

1. Introduction

It is a classical

problem

of

algebraic

number

theory

to decide if a number

field N of

_degree

n admits _power

_integrals

_bases,

that is

_integral

bases of the

~n-1 }

and,

if _yes, to determine all

_possible

_generators

of these bases. In case there exist _power

_integral

bases in

_N,

the field is called

monogeneous.

In a recent _paper Kable

_[9]

studied

_quartic

fields K with dihedral Galois

group,

having

power

integral

basis. These fields have a

_{unique quadratic}

subfield M. In the

_following

the discriminants of K and M will be denoted

by

DK

and

_DM,

_{respectively.}

The main theorem of

_[9]

_gives

necessary and sufficient conditions for K

containing

the subfield M to have a _power

integral

basis. The condition is

general

but does not allow us to settle the

_problem

of existence of _power

Manuscrit recu le 28 septembre 1999.

* Research _supported in _{part by} Grants 16975 and 25157 from the _Hungarian National Foun-dation for Scientific Research _by FKFP _0343/2000.

Lemma 1

_([9,

_Corollary

_1]).

Let K be a dihedral

_quartic

field

_containing

the

_{quadratic subfield}

M.

_If

K has a _power

integral basis, then,

with a

suitable choice

_of

_{sign, DK ±}

_4DM

is a _square.

In the

_special

case of dihedral

_quartic

fields with mixed

_signature

the

discriminant is

_negative

and the

_quadratic

subfield is real. Hence

Lemma 2

_([9,

_Corollary

_2]).

Let K be a rraixed dihedral

_quartic

field

con-taining

the

_quadratic

_subfield

M.

_If

K has a _power

_integral

basis,

then

4DM.

In

_particular

there are

only finitely

_many mixed dihedral

quartic

fields having

a _power

_integrals

basis and

_containing

a

_given

real

qua-dratic

_subfield.

Hence,

for a

_given

totally

real

_quadratic

field M we can enumerate all mixed dihedral

_quartic

fields K

_satisfying

the

_inequality

and then we can select those fields K

_containing

M as subfield and

having

power

integral

bases. This way we can determine all mixed dihedral

_quartic

fields K

_having

_power

_integral

bases and

_containing

the

_given

real

_quadratic

field M as a subfield. This is the _purpose of the

_present

_paper.

2. Power

_integral

bases in

_quartic

fields

For a

_{given quartic}

field K there are efficient methods for

determining

all

power

integral

bases,

cf.

Gaál,

Peth6 and Pohst

[3], [4], [5], [8].

A

general

fast method is described in

_{[6], [7].}

Assume that K is

_generated

_by

an

_{algebraic integer ~}

over

Q.

Let

f (t)

=

t4

+

alt3

+

a2t2

+ a3t + a4 E

Z[t]

be the minimal

polynomial of ~.

Let n =

I(~) =

(7LK :

Z[~]+)

be the index

_{of ~.}

We can

_represent

_any

_integer

a

in K in the form

with _{a, x, y,} z E Z where d E Z is a fixed common denominator. Note that

nld3.

Lemma 3

_([6,

Theorem

_2.1]).

The elerrtent a

_of

(2)

_generates

a _power

in-tegral

basis in K

_if

and

_{only if}

there is a solution

(u, v)

E

7G2

_of

the cubic

such that

_(x,

_y,

_z)

_of

₍₂₎

_satisfies

According

to the above remark the

_right

hand side of

₍₃₎

is an

integer.

If the form

_{F(u, v)}

is

_reducible,

then

₍₃₎

is trivial to

_solve,

otherwise it is a

cubic Thue

_equation.

For every solution

(u, v)

of

(3)

we have to determine the

correspond-ing

solutions

_{(x, y, z)}

of

_(4).

We follow the

_algorithm

of

_[7].

Denote

_by

(xQ, yQ, zQ) a

non-trivial solution of

If 0

_(the

other cases are treated

_similarly),

then there are rational

parameters

p, q, r such that

Since _any solution

_{(x, ~, z)}

of

₍₄₎

satisfies

_Qo(x,

_{y, z)}

= 0 we

_get

r(cip

+

c2q)

= ₊

c4pq +

C5q2

with

_integers

cl, ... , c5 which are

easily

calcu-lated.

_Multiply

₍₅₎

_by

_(cip

+

_c2q)

and use the above relation to eliminate

r. In

addition,

multiply

these

equations

with the square of the common

denominator of p, q to

get

integer relations,

and then divide the

_equations

by

gcd(p,

q)2.

This _way we obtain

where k &#x3E;

_0,

_c~~ are

_integers

and the

_parameters

_{p, q} E Z are

_{coprime. By}

[7]

k must divide

_{det(cj; ) /}

_{gcdf cij 13}

which is

_usually

a small

_{integer, allowing}

just

a few

possibilities

for l~. For each we substitute the

_{representations}

(6)

into

₍₄₎

to

_get

By

[7]

at least one of these

_equations

is a Thue

_equation

over the

origi-nal field K.

_Solving

this

_equation

we can determine _{p, q} and

_by

₍₆₎

the

corresponding

values z.

Thus in

_general

to determine all

_generators

of _power

_integral

bases in K

we have to solve a cubic and some

corresponding

quartic

Thue

_equations.

This can

easily

be done

using

the method of Bilu and Hanrot

[1]

which

is

_already

_implemented

in KASH

_[2].

Note that in our case for dihedral

3. The

_algorithm

input:

a real

_quadratic

field M

output:

all those mixed dihedral

_quartic

fields

_K,

that have _power

_integral

bases and contain M as a subfield. Moreover all

possible

_generators

of

power

integral

bases in K are

_computed.

1. Determine all those mixed

_quartic

_fields,

the discriminant of which satisfies

_(1).

(This

can be done

_by

_using

the tables

_{computed by}

KASH

~2).

Note

that

_by

M C K and

₍₁₎

we have

hence there are

relatively

few candidates for

K.)

2. Calculate the Galois groups and subfields of the candidate fields K.

We

_{keep only}

those

_quartic

fields K which have dihedral Galois _group and contain M as a subfield.

(This

is done

_again

_by

_using

_{KASH. )}

3. Determine all

_{possible generators}

of _power

_integral

bases of K. We

drop

the candidates K

_having

no _power

_integral

bases.

(Use

the methods of

_Gaál,

Peth6 and Pohst

_{[6], [7],}

_along

the lines of Section 2. As we noticed

_already

at the end of Section

_{2, equation}

(3)

is

_trivially

solved for dihedral fields. For the resolution of

_{(7), (8)}

we used

KASH.)

4. Numerical

_examples

To illustrate our

_algorithm

we

performed

computations

for M =

Q(V3-), Q(NF5).

The

_following

table contains the results of our

computa-tion. For each of these

_quadratic

fields M we list the discriminants

_DK

of

the mixed dihedral

_quartic

fields K

_containing

M as a subfield and

having

power

integral

bases. For each discriminant we

display

the minimal

poly-nomial

_{f (t)}

of the

_{generating element ~}

of

_K,

the common denominator d

of

₍₂₎

and the coordinates

_(x,

_y,

_z)

of the a in

(2)

generating

_power

integral

bases.

At some fields no solutions means that the field is a mixed dihedral

quartic

field

_containing

M as a

subfield,

but

_having

no _power

_integral

bases.

The method was

implemented

in

Maple,

the

quartic

Thue

_equations

(7),

(8)

were solved

by

KASH. The CPU time was a few minutes for _every

References

[1] Y. BILU, G. HANROT, Solving Thue equations of high degree. J. Number _Theory 60 (1996),

373-392.

[2] M. _DABERKOW, C. FIEKER, J. KLÜNERS, M. POHST, K. ROEGNER, K. WILDANGER, KANT

V4. J. Symbolic Comp. 24 _(1997), 267-283.

[3] I. _GAÁL, A. _PETHÖ, M. POHST, On the resolution of index form equations in biquadratic

number _fields, I. J. Number Theory 38 _(1991), 18-34.

[4] I. _GAÁL, A. _PETHÖ, M. POHST, On the resolution of index form equations in biquadratic number fields, II. J. Number Theory 38 _(1991), 35-51.

[5] I. _GAÁL, A. PETHÖ, M. _POHST, On the resolution of index form equations in biquadratic

number fields, III. The bicyclic biquadratic case. J. Number Theory 53 _(1995), 100-114.

[6] I. _GAÁL, A. PETHÖ, M. POHST, On the resolution of index form equations in _quartic number

fields. J. Symbolic Computation 16 _(1993), 563-584.

[7] I. _GAÁL, A. PETHÖ, M. POHST, Simultaneous representation of integers by a _{pair of ternary} quadratic forms - with an _application to index form equations in _quartic number fields.

J. Number Theory 57 _(1996), 90-104.

[8] I. _GAÁL, A. _PETHÖ, M. POHST, On the resolution of index form equations in dihedral number fields. J. Experimental Math. 3 _(1994), 245-254.

[9] A.C. _KABLE, Power integral bases in dihedral _quartic fields. J. Number Theory 76 _(1999),

120-129.

[10] L.C. _KAPPE, B. WARREN, An elementary test _for the Galois group of a quartic polynomial.

Amer. Math. Monthly 96 _(1989), 133-137.

István _GAÁL, Gábor NYUL

University of Debrecen Mathematical Institute H-4010 Debrecen Pf.12. Hungary E-mail : igaalOmath.klte.hu gnyuldragon.klte.hu