Polynomials whose Galois groups are Frobenius groups with prime order complement

J

OURNAL DE

T

HÉORIE DES

N

OMBRES DE

B

ORDEAUX

L

EONARDO

C

ANGELMI

Polynomials whose Galois groups are Frobenius

groups with prime order complement

Journal de Théorie des Nombres de Bordeaux, tome

6, n

o

_{2 (1994),}

p. 391-406

<http://www.numdam.org/item?id=JTNB_1994__6_2_391_0>

© Université Bordeaux 1, 1994, tous droits réservés.

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

Polynomials

whose Galois

_groups

are

Frobenius

_groups

with

prime

order

complement

by

LEONARDO CANGELMI

RÉSUME - On donne une caractérisation effective des _polynômes

irréduc-tibles de _{degré n} à coefficients entiers dont les _groupes de Galois _{sur Q} sont des _groupes de Frobenius avec _noyau d’ordre n et complément d’ordre

premier.

ABSTRACT - We _give an effective characterization theorem for _integral

monic irreducible _{polynomials f} of _{degree n} whose Galois _{groups over Q} are

Frobenius groups with kernel of order n and complement of _prime order.

0. Introduction

Recently,

there has been some interest in the

_problem

of

_giving

effec-tive characterizations of

_polynomials

of

_given

_degree

whose Galois groups

over Q

are Frobenius _groups of some

_particular

kind. Recall that a

Frobe-nius group can

always

be described as a semi-direct

product

N x H and

that N and H are

_respectively

called the

(Frobenius)

kernel and

(Frobe-nius)

complement

of the _{group; moreover,} the action

_by

_conjugation

of H

over N has to be

_{fixed-point-free,}

_~H)

has to divide

_{IN 1-}

1 and the group

can

always

be

represented

as a transitive

_permutation

_group on

~N~

ele-ments,

so that it is of

degree

Bruen,

Jensen and Yui

[BJY]

_proved

a characterization theorem for

_polynomials

of

_prime

_degree

_p whose

Ga-lois _groups are Frobenius _groups of

_degree

_p

_(in

this _case, the Frobenius

complements necessarily

are

_cyclic

of order

_dividing

_{p -}

1).

_{They posed}

the

_problem

of

_finding

_analogous

characterizations for

_polynomials

whose Galois _groups are Frobenius of

_prime

_power

_degree

or Frobenius in

_general.

In this _paper, we

_give

a characterization theorem for

_polynomials

of

arbitrary degree n

whose Galois groups are Frobenius _groups of

_{degree n}

and with

_complements

of

_prime

order _p, where p is forced to divide n - 1.

1991 Mathematics _{Subject Classification. Primary 12F10, 12Y05; Secondary 12F12,} 12-04.

Key words and phrases. Effective characterization of polynomials with _given Galois

groups, - Frobenius groups - - with prime order _complement.

It is worth

_noticing

that in the

_present

case the kernels are as

_general

as it

is

_{permitted by}

the

_properties

of a Frobenius _group, that

_is,

_they

have

_just

to be

_nilpotent;

while the

_complements

are restricted to be

_cyclic

of

_prime

degree.

The method we use is derived from one

by

Williamson

[Will],

who

characterized odd

degree polynomials

whose Galois groups are

generalized

dihedral _groups.

_Indeed,

these _groups are Frobenius with

complements

of order 2 and abelian

_kernels;

the fundamental observation is that his method does not

_require

the kernel to be

_abelian,

but

_just

the _group to be Frobenius.

_Moreover,

we make the characterization

_{completely effective,}

we discuss the

practical problems arising

and we

justify

theoretically

its

effectiveness

_referring

to "the least

_prime"

in Chebotarev

_density

theorem.

In Section

I,

we recall the definition of Frobenius _group and we show some

_interesting

_properties

of Frobenius _groups with

_complement

of

_prime

order: these can be described as semi-direct

products

N x

_Zp

satisfying

some

_simple

conditions.

In Section

_II,

we characterize monic irreducible

polynomials,

with

ra-tional

_integer

_{coefficients,}

of

_degree

_n, whose Galois groups

over Q

are

Frobenius _groups with kernel of order n and

_complement

of

_prime

order _p; that

_is,

_groups of the form N x

_Zp,

with

_INI

=

n, which are Frobenius.

In Section

_III,

we

point

out some effective conditions which are useful

in

_testing

whether a

polynomial

has Galois _group as above or

_not;

such

conditions involve the

_computation

of several resultant

_polynomials

and their factorization.

_Then,

we

apply

the characterization theorem to some

explicit

polynomials, showing

that their Galois groups are Frobenius _groups

with

_prime

order

_complement.

The author is

_grateful

to Prof. R. Dvornicich

_(University

of

_Pisa,

_Italy)

for some

_suggestions

and for several fruitful discussions.

Our notations are as follows.

If S is a

_set,

S)

denotes its

_cardinality.

If g

is an element of a _group which acts on a set

_{S, Fix g}

denotes the set of the elements of S fixed

_by

g.

For a _group

_G,

Aut

_(G)

denotes the _groups of

_{automorphisms}

of G. If A and B are two _groups, A x B denotes _any

_possible

semi-direct

_product

of them.

If n is a

positive integer, Zn, ~’n

and

Cn

denote

_respectively

the

_cyclic

_group

of order _n, a

_primitive

n-th root of

_unity

and the field if n is a

_prime

For a

polynomial f

E

_7G~X~,

_{spl ( f ), Gal ( f ),}

and

_{D f}

denote the

_splitting

field of

_{f ,}

the Galois _group of the

_splitting

field and the discriminant of

_{f .}

For _any number field

_{L, DL}

denotes the discriminant of L

_{over Q}

and

OL

denotes the

_ring

of

_integral

elements of L.

For _any finite extension of number fields K

_(also

written as

L/K)

and

any

f

E

_L(X~,

_{NL~K ( f )}

denotes the norm

of f over

K. For _any finite normal

extension of number fields

_L/K,

Gal

_(L/K)

denotes its Galois group.

If k is a field or a

polynomial

_ring,

and

f (X)

and

g(X)

are two

_polynomials

in

_{k [X],}

denotes the resultant of

_{f and g}

with

_respect

to the indeterminate X.

Other notations will be recalled and

_adopted

in the course of the _paper.

I. Frobenius _groups 1.0. General results

DEFINITION I.0.1. A

_permutation

group G on a set S2 with

_In

= n is said

to be a _group of

degree

n.

DEFINITION I.0.2. A _{transitive group} G of

_{degree n}

is said to be a

_regular

group

if,

and

only

if,

IFix91

=

_0,

for

all g

_E

GB{1}.

DEFINITION I.0.3. A transitive group G of

degree n

is said to be a

Frobe-nius group

if,

and

_{only if,}

_1,

for

_{all g}

E and

_IFix91

_{= 1,}

for

some 9 E

_GB{1}.

It is trivial to see that a transitive group G of

_{degree n}

is

regular if,

and

only if,

~G~

= _n. On the other

hand,

_a transitive group G of

degree

_n is _a

Frobenius _group

_if,

and

_{only if,}

it has a non trivial

_subgroup

_H,

of index _n,

such that H n H9 =

{1},

for

all g

_E

GBH

(anyone

of the stabilizers _can be

taken as

H).

This

_property

allows to deduce a _very

_strong

theorem about

the structure of Frobenius _groups.

THEOREM 1.0.4

_~FROBENIUS~.

Let G be a Frobenius _group

of degree

n and

H be a

_subgroup

(of

index

n)

such that H fl H9 =

111,

for

all _{g E}

GBH.

Then the subset

is a normal

_subgroup

(of

order

n),

such that G = NH and N _n H =

{1}.

Since the

_conjugates

of H are

_exactly

the stabilizers of the

_points

of

52,

the elements of

_NB{1}

are

_precisely

those in G with no fixed

_points.

N

is called the

_(Frobenius)

kernel of

_G,

while H is said to be a

(Frobenius)

complement

of G. Note that N turns out to be a

_regular

_group of

_{degree n}

and that G can be written as a semi-direct

product, namely

G = N x H.

Furthermore,

G satisfies some other

_strong

_necessary

_conditions,

as it is

stated in the

_{following proposition.}

PROPOSITION 1.0.5. Let G = N x H be a Frobenius _group

of degree

_n,

= n and

_IHI

= h. Then the

following

conditions hold:

a) CG (H) =

CH (H)

b)

H is

_core-free

_{(i. e., it}

does not contain non-trivial normal

_subgroups

of

G).

c) h I n - 1.

d) (Thompson)

N is

_nilpotent.

Proof. See,

e.g.,

[Rob].

1.1. Frobenius _groups with

_prime-order

_complement

Let N be a _group of order n

having

an

automorphism 0

of

_prime

order _p

and let

_Zp

denote the

_cyclic

_group of order _p. We can construct the semi-direct

_product

G = N _xo

Zp:

if

Zp = (T),

then we set Tx =

8(x)r,

for

all x E N.

_Then,

we _may

verify

whether G is Frobenius

by

means of

several different conditions.

PROPOSITION I.1.1. Let G = N

Zp,

n = and

p

f

n. Then the

following

conditions are

_equivalent:

a)

G is a Frobenius group

of degree

n.

b)

=

Zp.

c)

d)

For _{any x} E

_N,

there exists

_exactly

one element

y E N

such that

O

e)

e(p-l)i(x)

=

1,

for

all _{x E} N and

for

all i =

1, ... , p -

1.

f) Any

element in

GBN

has order _p.

g)

G has

_exactly

n

Sylow p-subgroups.

Proof.

a)~b):

see

Proposition

1.0.5

(a).

b)=&#x3E;c):

by

the _very definition

of G,

we have that Fix 0 =

CN(T)

=

hence,

F1x 0 =

(1) ,

because the

only

elements

commuting

with r

C)==&#x3E;d):

if

_x-19(x)

=

y-1B(y),

with _x and _{y in}

N,

then

yx-1

=

0(yx-1)

and therefore = 1.

d)~e):

let y

=

z0 (z) ...

then

0(y)

= =

y x

We _{may write x in} the form

_z-’O(z)

and hence we

_get

zyz-i;

by

the

_uniqueness

of such

_expression,

we

_get

zyz-1

=

_1,

hence _y = 1. This _proves the

implication

in the _case i = 1. For the

general

case, note that condition

_(e)

for i = 1

implies

condition

(c);

on the other

_hand,

_by

the

_primeness

of _p, we have Fix 0 =

Fix 0’.

So 0’

is an

_automorphism

of N with the same

_properties

as 0 and

therefore the assertion holds for all i.

e)#f):

any element in GBN can be written in the form

_xTi,

for some x E N and some i E

_{{1, ... , p-1}:}

then

_(xTi)P

=

_1,

since

(xTi)P

=

...

f)~g):

G contains

_exactly

_IGBNI

=

n(p -1)

elements of order

p and

they

give

rise to

_{exactly n subgroups}

of order _p.

g)==&#x3E;a):

Zp

has

_exactly

n

_conjugates

and _anyone of them can be written

as

_Z~,

for some x E

_N;

_{if g}

E

_GBZp,

then

_Zj

=

Zp

for some x E hence

_{Z~ ,}

and therefore

_Zp

f1

_ZP

_{= { 1 } .}

Condition

_(c)

of the

_{previous proposition}

_says that the

_{automorphism 0}

is

_{fixed-point-free.}

A Frobenius group with kernel N and

complement

Zp

will be denoted

by

Fr

_{(N, p).}

It is

_plain

that such a _group satisfies the

_following

_properties,

besides the ones claimed in

Proposition

I.1.1

(cf.

Proposition

1.0.5):

- ord

_(B)

_{= p.} - - N is

_nilpotent.

- p

I IAut

In

_particular,

N is the direct

_product

of its

_{Sylow subgroups: if q}

_{n and}

Nq

is the

_{Sylow q-subgroup}

of

_N,

then

_81Nq

is a

fixed-point-free

automor-phism

of

_NQ

of order _p.

_Conversely,

_given

a

fixed-point-free

_automorphism

of order _p for each of the

_{Sylow subgroups}

of

_N,

it is determined a

fixed-point-free automorphism

of order _p of N.

_Therefore,

the

_study

of groups as

Fr

_{(N, p)}

is reduced to that of Frobenius _groups with

_Zp

as

_complement

and a _q-group as

kernel;

and this is

equivalent

to the

_study

of

_{fixed-point-free}

automorphisms

of order p of q-groups, with

q ~

p.

I.2.

_Cycle

structure of elements of Fr

_{(N, p)}

In

_general,

_given

a

_permutation

_group G of

_{degree n}

and

_assigned

a

representation

of G as a

subgroup

S of

Sn,

_any

_{element g}

E G

_gives

rise to

a

_partition

7r(g)

of n: in

fact, g

can be written as a

_permutation

of

_degree

n

(via

the

_given

_{representation}

of

_G),

and as such it induces a

_partition

of n

given

by

the

_lengths

of the

_cycles

in the

_expression

_{of g as product}

of

_disjoint

cycles.

Note that for _any

_permutation

_{of f 1.... , n~,}

the new identification

of G with a

subgroup

of

Sn

is

_given

by

a

_conjugate

of

,S’;

hence the

_partition

remains

_unchanged.

We _say that an

_{element g}

in a _group G of

_{degree n}

is of

_type

_7r, where 7r

is a

_{given partition}

of _n,

if,

and

_{only if,}

= _7r, for a

_given

_{representation}

of G as a

_permutation

_group of

_degree

n. We will write

_"partitions"

in

multiplicative form;

that

_is,

_{nil - n22 - - -}

will denote the

_partition

of

_E

ein2

given

by

el times nl, e2 times n2, ....

PROPOSITION I.2.1. Let G = and

_represent

G as _a

_permutation

group

of degree

n. Then the

_{partition types}

induced

_by

G are the

_following:

-

for

the

_{identity of G.}

-

1 -

_for

each element in

_GBN.

-

n1 - - - nr, with some 2 such that _p

~’

_{nj and}

I

_n,

for

each

element in

Proof.

Each

_{element g}

in GBN has order _p and so it is

product

of

disjoint

cycles

of

_length

_p or

_1;

_moreover, since G is Frobenius of

_degree

_{n, any} element can fix at most one

_point.

Hence the

_{only possibility}

for is to

be

_equal

to 1 - For the elements in note that

_they

fix no

points

and that

_{p t}

n .

II. Characterization of

_polynomials

of

_{degree n}

with as Galois _groups

ILO. Factorization

_types

of

_polynomials

Let

_{f E}

_Z[X]

be a monic

polynomial

of

_{degree n}

and let 7r be a

_partition

of n. We _say that

f

is of

_type

7r

if,

and

only if,

the

_partition

of n

_given

by

the

_degrees

of the irreducible factors of

_f

is 7r. We _say that

f mod q

is of

type

7r,

where q

is a rational

_{prime, if,}

and

_{only if,}

the

_partition

of n

_given

by

the

_degrees

of the irreducible factors

_{mod q}

of

_f

is 7r.

Again,

if E _{is any}

number

_field,

we _say that

_f

over E is of

_type

7r

_if,

and

_{only if,}

the

_partition

of n

_given

_by

the

_degrees

of the irreducible factors over E of

f

is 7r.

We have the

_{following deep}

relations between the factorization

_types

of

_f

and the

_cycle

structure of the elements of Gal

_{( f )}

_(this

can be

faithfully

represented

as a

_permutation

_group of

degree

_n, on the set of roots of

_{f ,}

and _any

_permutation

of the roots reflects on

taking

a

conjugate

of the

given

group

representing

Gal ( f )

in

Sn).

LEMMA II.0.1. Let

_f

E

_{Z[X] be}

a monic

polynomial of degree n

and let 7r

be a

partition

of

n.

Let q

denote _any rational

prime

such that

q f

D f

and

Let G =

Gal ( f ) . Put L = spl( f ) .

a) (Dedekind)

If f mod q is of

type

then G contains an elements

of

type

7r.

b)

(Frobenius)

If

G contains elerraents

_of

_types

then the

_{density of}

q’s

such that

_f

mod

_{q is}

_of

_type

7r is

_positive.

c)

(Chebotarev)

The

_{density of q’s}

such that

_f

mod _{q is}

_of

_type

7r is

equal

to the

_proportion

_of

elements in G

_of

_type

7r.

d) (Lagarias,

Odlyzko

et

_al.)

_If

G contains elements

_of

_type

then there exists q such that

_f

mod _{q is}

_of

_type

7r and _q

(DL)C,

where

C is an

effectively computable

absolute

_constant;

assuming

the

Gen-eralized Riemann

_{Hypothesis for}

_L,

one has

_q

70(log

DL) 2.

Proof.

a)

See,

e.g.,

[Jac,

pp.

302-304].

b)

See

_[Frob].

c)

See,

e.g.,

[Lang,

pp.

168-170].

d)

See

_[LMO]

and

_[Oes].

11.1. Characterization theorem

Let

_L/Q

be a normal extension and let K be a subfield of L such that

L/K

has

_degree

_n,

_K/Q

is a

cyclic

extension of

prime

degree p

and

Put G = Gal

(L/Q)

and N = Gal

(L/K);

then,

since

(p, n)

=

1,

L/Q

splits

at K and G =

ZP.

Moreover, K

is the

_unique

subfield of L of

_degree

p

over

Q,

because N is the

unique

subgroup

of G of order n. The

product

N x 0

Zp

is direct

_if,

and

_{only if,}

Fix 0 =

N;

if this is not the _case, then

Zp

is not normal and therefore is core-free.

PROPOSITION II.1.0. Let

_L/Q

be a normal extensions

of degree

_pn, with

Gal

_(L/Q)

= N

Zp and p t n.

Then the

product

is not

direct if,

and

only

if,

there monic irreducible

_polynomial

_{f ,}

with

_integer

_coefficients

and

_{of degree}

_n, such that L =

spl

( f ) .

Proof.

If the

_product

is not

_direct,

_Zp

is core-free and the minimal

polyno-mial of an

_{integer primitive}

element of satisfies the

_required

proper-ties. If the

_product

is

_direct,

L contains a

unique

field of

degree

n over

Q,

In

_general,

a normal extension

L/Q

whose Galois group is a non-direct

product

as above

might

also be

_given

as the

splitting

field of an irreducible

polynomial

of

_degree

not

_equal

to n; anyway, the

previous proposition

says

that we can

always

think of it as the

splitting

field of a

polynomial

of

degree

n.

For the rest

_of

this

_subsection,

_f

E

_Z[X]

will denote a rraonic irreducible

polynomial of degree n

and we will

_put

G = Gal

( f )

and L =

spl

( f ).

If G = Fr

(N, p),

f

is irreducible and normal _over

LN,

since N is

regular;

moreover, from

Proposition

1.2.1 and Lemma

11.0.1,

we know that there

exists a rational

_prime

_q such that

f mod q

is of

_type

1 - Such

conditions turn out to be also

_sufficient,

as we are now

_going

to show in

two

_steps.

PROPOSITION II.1.1. Let _p be a rational

_prime

such that p

{

n. Then the

following

conditions are

_equivalent:

-

G = N _~ae

Zp, IN

= n and

f is

irreducible over

L~.

-

p ~

I

[L : Q]

and there exists a

_cyclic

extension k

_{of Q of degree}

_p, such that

_f

is irreducible and normal over k.

Proof.

The first condition

_{trivially implies}

the other one,

by taking

k =

L~’.

Conversely,

since

_{deg f ~}

_{~L : Q]}

and

_{(p, n)}

=

1,

_we have

pn ~

I [L : Q].

Let

E be the

_splitting

field of

f

over

_k;

_by

the

_hypothesis,

[E : k~ = n and

therefore

_{[E : Q]}

= _pn. Since

L C E,

_we

_get

L =

E; hence,

k C L and

G = N _xe

Zp,

where N = Gal

(L/k).

PROPOSITION 11.1.2. Let

_{pin -}

1 and let the

_equivalent

conditions

_of

Proposition

11.1.1 hold. Put K =

LN.

_If

there _{exists a} rational

_prime

q,

unramified in

L,

such that

_f

mod q is

of

type

then G = Fr

(N, p);

rraoreover, q is inert in K and

qOK splits completely

in L.

Proof.

Since

_{[K : Q]}

= _p, for _any rational

prime

_q, unramified in

L,

there

are

_just

two

_{possibilities: either q}

is inert in K or it

_{splits completely}

in K. If the latter is the case, then

qOK

=

C~1 ~ ..

C~~

and =

1;

hence, reducing f

mod

_Qi

is

_equivalent

to

_{reducing f}

mod q. In the case

of the

_hypothesis,

we would have that

f

mod

Qi

is of

_type

1 -

and therefore

_(see

Lemma 11.0.1

_(a),

which

_applies

also to number fields-cf.

_[vdW,

_pp.

_190-191])

Gal

_(L/K)

would contain an

_element,

not

_equal

to the

_identity,

which fixes a root of

_{f ;}

but this is

_impossible,

since N =

Gal

_(L/K)

is

_regular

on the roots of

_{f .}

This shows that _{q is}

_necessarily

inert in K.

So,

deg(qOK/q)

= _p and therefore

lF9p.

Since the

factor-ization of

_f

over

_IFq

is of

_type

1 . the factorization of

f over

_Fgp,

and hence over is of

_type

1’~.

Being f

irreducible and normal over

K,

L =

K(a)

for _any root a of

_{f ;}

_then,

_{splits completely}

in

_L,

Being

= _p

and q

unramified in

L,

the

decomposition

_group

of

_Qi,

_GQ,,

is

_cyclic

of order p, i.e.

GQ, cr Zp.

Putting

F =

Q(a),

where a is any fixed root of

_{f ,}

from the factorization of

_{f mod q}

we

_get

Bi ..

B,

with

_deg(Bh/q)

= _p and =

1,

for

h =

1, ... ,

(n -

1)/p.

Comparing

the factorizations of

qOF

and of

qOL,

we see that there are

_exactly

_p

_primes

of L above

Bh ,

while there is

_exactly

one

_prime

above

,Ci,

_say

Q.

Therefore F D

LGQ

and

F 1J

LGQ’,

for 6’ !

This

_implies

F =

LGQ

and

L GQ 0

LGQ’,

and _so

GQ , 0 G Q, -

Therefore,

G

contains n distinct

_{cyclic subgroups}

of order _p, the

_{highest possible}

_number,

hence G has n

_{Sylow p-subgroups}

and it is Frobenius

_{by Proposition}

I.1.1.

By

the same

_technique

used in the

_proof

of the

_{previous proposition,}

we can show also the

following fact,

_concerning

the kind of

_{primes q}

which

_give

rise to factorizations of

_f

mod q of

type

1 .

PROPOSITION II.1.3. Let G = Fr

(N, p),

with

INI = n and p I n - 1.

_Then,

for

any rational

prime

q,

unramified

in

L,

f mod q is of

type

1.

and

_{only if,}

_{q is inert in} K and

_qOK

_splits

_completely

irc L.

Proof.

Omitted.

We are now in a

_position

to

_give

the announced

_{characterization, just}

putting together

all the above results.

THEOREM II.1.4. Let _p be a rational

_prime

dividing

n - 1. Then G =

Fr (N, p) if,

and

_{only if,}

the

_following

conditions hold:

1.

_{p I [L : Q].}

2. There exists an extension

k/Q,

cyclic of degree

_p, such that

f

is

irreducible and normal over k.

3. There exists a rational

_prime

_q,

unramified

in

L,

such that

_f

mod q

is

_of

_type

1 -

_{p(n - 1) /p.}

We will see in the next section how this characterization can be made

III. Effectiveness of the characterization

111.0. Resultants and factorization of

_polynomials

over number fields

Let K =

k((3)

be _a number

field,

where k is a subfield of K and

_Q

is a fixed root of a

_given

a monic irreducible

polynomial g

E

_k(X~,

of

degree

m; let

(31,... ,13m

denote the roots of g. Let

f

E

_K~X~

be a monic

square-free polynomial

of

_{degree n}

and al, ... , a~, be the roots of

_{f ;}

we

may write

f

as a

polynomial

in

k[X, (3],

f

=

f (X, (3).

We _{want to} consider

the factorization

_type

of

_f

over

K;

_suppose that such a factorization is

/i’ " ft.

For _any

_polynomial

E

_K[X]

and _any rational

_integer

s, let

N(s,

h,

g)

stand for the norm over k of the shifted

polynomial

h(X -

s (3,

{3) ,

that is for

_(3)).

_Referring

to the factorization

_algorithm

of

_polynomials

over number fields due to

_Trager

_(see,

_also,

_[vdW],

pp.

136-137),

we can affirme the

following:

i)

there are

only finitely

_many rational

integers s

such that

N(s,

f,

g)

is not

_square-free;

ii)

for _any rational

_{integer s}

such that

_N(s,

_f,g)

is

_square-free,

the factorization over k of

_{N(s, f,}

_g)

is

_N(s,

_fl,

_{g) ... N(s,}

_ft,

_g).

Thus,

each irreducible factor of

_{f over}

of

_degree

d _maps to an

irre-ducible factor of

_{N(s, f, g)}

over k of

degree

nd,

and vice versa.

Moreover,

we are able to calculate the norms above

by

means of the

function

_resultant,

which is available on _any

_computer

_algebra

_system;

in

fact,

the

_{following equalities}

hold

_(the

last one _up to a

sign):

In

_particular,

when

_f

E

_k[X]

is irreducible over

k,

we _may

study

the

factorization

_type

of

_f

over the field

k(a),

where a is a fixed root of

_{f .}

In such a _case, one of the factors of

f

is X - a

and,

since we are

obviously

interested in the other

_factors,

we define the

following polynomial

which is a monic

_polynomial

in

_k[X]

of

_degree

_{n(n - 1)}

and it is a linear

resolvent of

_{f when s 0}

_{0,1. Then,}

we claim that

_f

is normal

_{over k,}

i.e.

_spl

_{( f )}

=

k(a)

for

type

In

_fact,

_f

is normal over k

_if,

and

_only

_if,

_f

factors over

k(a)

as

the

_product

of n linear

_polynomials,

which is

_equivalent

to

_{N(s, f )}

_being

of

_type

nn over

k,

hence to

_{R(s, f )}

_being

of

_type

over k.

III.1. Factorizations of

_{R(s, f )}

For this subsection and

_for

the next one, assume

f

E

_Z[X]

to be a monic

irreducible

_polynornial

_{of degree}

_n, such that Gal

_{( f )}

= Fr

_{(N, p),}

with

_{IN I}

n, and

Put

spl ( f )

= L and K =

L~’.

Fix a root a of

f

and let a’ be _any other root of

f .

How does

f

factor over

_Q(a)?

_Firstly,

note that

_{[L : Q(a)]}

=

p and therefore any irreducible

factor of

_f

over

Q(a)

_may be of

degree

either 1 or _p.

Then,

since Gal

( f )

=

Fr

_{(N, p),}

we know that there exists a rational

prime

_q, unramified in

L,

such that

_f

mod _q is of

_type

1

_{by Proposition}

_II.1.3,

_{q is inert in}

K and

_{qO K splits completely}

in L.

_{If qOL}

=

61 " ’

then,

proceeding

_as

in the

_proof

of

_Proposition

_11.1.2,

we find that the n

Sylow p-subgroups

of

G = Gal

( f )

are the

_{decomposition}

_groups of the

primes

of L over _q,

and that the

_respective

_{decomposition}

fields are

_precisely

the extensions

of Q by

the roots of

_{f ,}

i.e.

L GQi =

_{Q(ai) (i}

=

1, ... , n).

This

implies

Q(a) #

Q(a’)

and so

a:’ ft

Q(a).

Hence,

the minimal

polynomial

of a’ over

Q(a)

has

_{degree equal}

to p.

Thus,

we have

_proved

the

following

fact.

PROPOSITION III.1.1. Let

_f

be as above and let a be a root

_{of f.}

Then

_f

over

Q(a)

is

of

_type

1 -

and,

for

_any rational

integer

s such that

R(s, f ) is

square-free,

R(s,

f )

over Q

is

_of

_type

Since

_f

is irreducible and normal over

K,

from the last remark in the

previous subsection,

we have also the

_following

_result,

_involving

the

factor-ization of

_R(s,

_{f )}

over K.

PROPOSITION III.1.2. Let

_f

be as above.

Then,

for

_any rational

_integer

s

such that

_R(s,

_{f )}

is

_square-free,

_{R(s, f)}

over K is

_of

_type

Since we are

_just

interested in the factorization

_type

of

R(s, f )

over K

and not in the actual irreducible

_factors,

we can

apply

_again

the method

described in the

_{previous subsection,}

_getting

the

_following.

PROPOSITION III.1.3. Let

_{f as}

above

_{and g}

E

_Z[X]

be a monic irreducible

polynomial

such that K =

Q(Q),

where

Q

is a root

_of

_g.

_Then,

_for

_any

rational

_integer

t such that

_N(t,

_f,

_g)

is

_{square-free, N(t,}

_f,

_g)

is irreducible

for

any rational

integer

t such that

N(t,

R,

g)

is

square-free,

N(t,

R,

g)

over

Q is

of

type

(np)n-1.

III.2. Determination of K

Recall that K is

_cyclic

of

_prime

_degree

_p over

Q. Thus, being

abelian over

the

_rationals,

_by

Kronecker-Weber

_theorem,

I~’ is included in a

cyclotomic

field

_Cl

_(where

_Cl

=

Q((),

with C

_a

primitive

1-th _root of

unity),

for _some

positive

integer

such

_{that q}

_f

_DK

_{implies q}

_{

_DCl’

for _any rational

_prime

_q.

Assuming

l to be minimal with

_respect

to this

_property,

its factorization

is to be of the kind

_{p2aql ... qs,}

where a E

_{10, 1}}

and _{qj ==} 1 mod _p, for

j

=

1, ... ,

_s.

Moreover,

since

DK ~

I DCl’

_we have

Now observe that

_f

is irreducible and normal over

_{K; hence,}

L =

K(a),

for _any root a of

f .

From

this,

it follows that and

therefore,

if q

l,

then

_q

_{I Df.}

_Seeing

the factorization of

_l,

we deduce

that l pD f.

Thus,

if the factorization of

_{D f}

is

with eo 2:

0,

ei 2:

1,

f~ &#x3E;

1,

1 mod _p 1 mod _p, for i =

_{1, ... ,}

_r

and j

=

1, ... , t, putting

we

have l

m.

Therefore,

K C

_Cl

C

_Cm

and K can be determined as one of the

_cyclic

subfields of

_degree

p

over Q

of

Cm.

These are

finitely

_many, since

_they

correspond

to the

_subgroups

of index _p of Gal

_{(7~/m7~) *;}

_indeed,

their number is

_precisely

_{1) I (p -}

_1).

111.3. Effective version of the characterization theorem

We are now able to

_give

an effective version of Theorem 11.1.4. For

simplicity,

we assume

GRH;

_so, we can use the

_strongest

bound

_given

in

Lemma II.0.1. This is not so

restrictive,

as it is shown

_by

all tested

exam-ples,

where the least

_{prime q}

is much smaller than the one

_given

theoreti-cally,

even under such

_assumption.

THEOREM III.3.1. Let

_f

E

_Z[X]

be a monic irreducible

_{polynomial of}

de-gree n and

Put

G = Gal

( f )

and L

= spl

( f ).

Let _p be a rational

prime

such

that

_{pin -}

1. Assume GRH.

If

G = Fr

_{(N, p),}

with

_)N)

=

n,

then, putting

K =

L~’

and R =

R(s, f ),

for

sorrte rational

_integer

s such that

R(s, f )

is

square-free,

we have:

i. R is

_of

_type

_(np)(n-l)/p.

ii. K is one

of

the

cyclic subfields of degree

_p

over Q

of cm,

where m

is

_given

in Section

_111.2;

moreover,

f

is irreducible over K and R

over K is

_of

_{type nn-1.}

iii. There exists a rational

_prime

_q, not

dividing

D f,

such that

_q

and

_f

mod

_{q is}

_of

_type

1.

That such conditions are sufficient for G = Fr

(N, p)

_was

already proved

in Theorem

_11.1.4;

the

_point

is that now we have an effective _way to

_verify

the conditions

_given

in that theorem. III.4.

_Examples

Although

the

_given

characterization determines if the Galois _group of a

polynomial

of

_{degree n}

is Frobenius with kernel of order n and

complement

of

_prime

order _p

_{dividing n -}

_1,

a

_general

method for

_constructing

such

polynomials

is not

_known,

_apart

from few cases.

Moreover,

in some cases a

theoretical method may be

outlined,

but the

computations

involved exceed

the

_capacity

of the

_computer

machines available to the

_author,

so that we are not able to exhibit many

polynomials

of the

required

degree

and with

the

_required

Frobenius Galois _group.

Therefore,

to

_give

some

examples

of how our method

applies,

we must refer to

_polynomials

found in the

literature,

and somehow

_manipulate

them.

Example

1. The

_{simplest example}

of a

polynomial

considered

by

our

method is a

polynomial

of

degree

4 whose Galois _{group is} the

alternating

group of

degree

4,

x

_{Z2, 3).}

Such

_polynomials

are _easy to construct.

_Take,

for

_example,

the

_following

one:

We will _prove that its Galois _{group is}

_A4,

_verifying

the conditions

_given

in

Theorem 111.3.1 and

_using

also

_Proposition

III.1.3.

-

The smallest

_prime

q for which

f,

mod q factors as 1 -

32 is

3. -

The resolvent R =

R(-1, fl)

is

square-free

and it is irreducible

over

Q.

- The discriminant of the

polynomial

is

_21272132

so the

_possible

cubic

included in

_C7;

a

_primitive

element for such cubic

field,

which we

call

_.K,

is a root of the

_polynomial

We

_compute

_fl,g)

which turns out to be

_square-free

and it is

irreducible.

_Then,

we

_compute

N(1, R, g)

and we

verify

that it is

square-free

and of

_type

123.

Hence,

we claim that the Galois _group of

f,

is

A4

and that the

_unique

cubic field included in its

_splitting

field is the

_unique

cubic field included

in

_C7,

that is K.

Example

2.

_Bruen,

Jensen and Yui

_[BJY]

constructed a

family

of

ra-tional

_{integral polynomials}

of

_degree

7 whose Galois groups

over Q

are

Frobenius _groups of order

_21,

that _{is groups} as Fr

(Z7, 3).

From this

family,

we have taken the

_following

_polynomial

We

_apply

our characterization theorem to this

_polynomial.

- We factor

f2 mod q

for some rational

_primes

_{q, in} order to find a

factorization of

_type

1 ~

32 -

should we find a factorization

_type

not

equal

to

_17,

1.3 2

or

_7,

we would be sure that Gal

(f2) 0

Fr

(Z7, 3)

(see

Proposition

I.2.1).

The

_prime

3

_gives

the

_required

factorization.

- We

compute

the resolvent R =

R(-1, f2),

which _{turns out to} be

square-free

and factors

_{over Q}

into 2 irreducible

_polynomials

of

degree

21, Rl

and

_R2,

as we

expected.

-

The discriminant of

_f2

is

_equal

to

224710.

Therefore,

the

_unique

possible

cubic field to consider is

_K,

the cubic field included in

_C7.

The norm

N(1,

f2, g)

is

_square-free

and it is irreducible. The norm

=

N(1,

Ri,

g)N(1,

R2,

g)

is

square-free

_as well and it is

of

_type

216.

Hence,

we have verified that the Galois _group of

_f2

is Fr

_{(Z7, 3)}

and we

have shown that the cubic field included in its

_splitting

field is K.

Example

3. To find other

_{explicit polynomials,}

we refer to a

_proposition

by

Williamson. If _gi and _g2 are two monic

_polynomials

in

_k[X],

where k

is any number

_field,

let

_{S(gl, g2)}

denote the

_polynomial

whose roots are all

the sums of one root of _gi and one root of _g2; we can

easily

_compute

such

polynomial by

the

_following

formula:

PROPOSITION III.4.1. Let 91 and 92 be irreducible monic

polynomials

over a number

field

k and let

El

and

E2

be their

_respective

splitting fields

over k.

If

E1

and

_E2

are

linearly disjoint

over

k,

then the

polynomial

S(gl, 92)

is

irreducible over k and its

_{splitting field}

over k is

_ElE2.

Proof.

See

_[Will]

-We will

_apply

such a result to the

_{polynomials f,}

and

_f2,

which are

irreducible over K. We

_compute

the

polynomial

f3(X)

=

S(fl, f2)(X),

which is

_equal

to

Since the

_splitting

fields over K of

_f,

and

_f2

have

_relatively

_prime

_degrees

over

K,

they

are

linearly disjoint

over

K,

so that

f3

is irreducible over K

and the Galois group of its

splitting

field over K is

_Z2

x

_Z2

x

_Z7:

_being

the

_degree

of

_f3

_equal

to

_28,

we deduce that

f3

is also normal over K. It is

also obvious that the

_degree

of the

_splitting

field of

_{f over Q}

is a

multiple

of

_3,

since such field includes K.

_So,

conditions 1 and 2 of Theorem 11.1.4

are satisfied and the

_only

left verification is about the factorization

_type

of

f3

mod _q: the smallest

_{prime q}

which

_yields

the

_type

1’

39

is 11.

Hence,

Gal (/3)

=

Fr(Z2

_x

Z2

_x

Z7, 3)

and the cubic field included _in

spl ( f3) is K.

Example

4. We

_proceed

as in the

_previous

case. From the mentioned

family

of

_{polynomials by}

_Bruen,

Jensen and

_Yui,

we take another

polyno-mial,

whose

_splitting

field

_again

includes K as above. We construct the

polyno-mial

_fs(X)

=

S( f2,

f4)(X)

and we

_verify

that it is irreducible over

Q.

This

implies

that the

_splitting

field of

_f2

and

_/4

are

different,

hence that

_they

are

disjoint

over K and that the Galois group of

f5

over K is

Z7

x

_Z7.

_As in the

previous

example,

conditions 1 and 2 of Theorem 11.1.4 are

plainly

satisfied.

Then,

we find that 11 is the smallest

prime q

such that the factorization

type

of

_f5

mod q is 1 .

316.

Therefore,

Gal(/5) = Fr(Z7

x

_{Z7, 3)}

_{and the cubic field included} in

spl ( f5)

is K. We do not

_report

_f5,

because its

_greatest

coefficients are

48-digit

numbers.

REFERENCES

[BJY]

A. A. _Bruen, C. U. _Jensen, N. _{Yui, Polynomials} with _{Frobenius groups} of prime

deegre as Galois groups II, J. Number Theory 24 (1986), 305-359.

[Frob]

G. _Frobenius, Über Beziehungen zwischen den Primidealen eines _{algebraischen}

Körpers und den Substitution seiner _Gruppe, S. B. Akad. Wiss. Berlin _(1896), 689-705.

[Jac]

N. _Jacobson, Basic _{algebra I,} 2nd ed., Freeman, New _York, 1985.

[Lang]

S. _{Lang, Algebraic} number theory, GTM _{110, Springer-Verlag,} New _York, 1986.

[LMO]

J. C. _Lagarias, H. L. _Montgomery, A. M. _{Odlyzko, A} bound _for the least _prime ideal in the Chebotarev density theorem, Invent. Math. 54

_(1979),

271-296.

[Oes]

J. Oesterlé, Versions _{effectives du} théorème de Chebotarev sous _{l’hypothèse} de Riemann _{généralisé, Astérisque} 61 _(1979), 165-167.

[Rob]

D. J. S. _Robinson, A course in the theory of groups, GTM 80, Springer-Verlag,

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L. _Cangelmi

Dipartimento di Metodi e Modelli Matematici per le Scienze Applicate

Universita di Roma "La _Sapienza" Via A. _Scarpa 10

00161 Roma, Italy