On the annihilating ideal for trace forms

J

OURNAL DE

T

HÉORIE DES

N

OMBRES DE

B

ORDEAUX

M

ARTIN

E

PKENHANS

On the annihilating ideal for trace forms

Journal de Théorie des Nombres de Bordeaux, tome

15, n

o

_{1 (2003),}

p. 115-124

<http://www.numdam.org/item?id=JTNB_2003__15_1_115_0>

© Université Bordeaux 1, 2003, tous droits réservés.

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

On

the

_annihilating

ideal for

trace

forms

par MARTIN EPKENHANS

RÉSUMÉ.

Npus

donnons

_{plusieurs exemples}

de familles de formes trace dont l’idéal annulateur dans

_Z[X]

est

_principal.

Nous

mon-trons aussi

_{qu’en général,}

cet idéal n’est _pas

_principal.

ABSTRACT. We

_give

several

_examples

of classes of trace forms for which the ideal of

_{annihilating polynomials}

is

_principal.

We prove, that in

_general,

the

_annihilating

ideal is not a

_principal

ideal.

1. Introduction

My

talk

_given

at the 20th Journees

_{Arithmétiques}

at

_Limoges

in 1997 concludes with a

_question

on the

_injectivity

of a certain _map defined in

the context of Burnside

_rings

and trace forms. Now we are able to

_give

an affirmative answer. Theorem 6 enables us to reduce

_questions

on trace

forms to

_{corresponding}

_questions

of trace forms of

_2-groups.

Let K be a field of characteristic different from 2. Since the Witt

_ring

W(K)

over K is an

_{integral ring}

we _may consider

polynomials

in

Z[X]

evaluated at an

_{element 0}

of

_W(K).

We say a

polynomial

p(X)

E

_Z[X]

annihilates 0

= 0 in

W (K).

Definition 1. Let M be any class of

quadratic

forms. Then the

annihilat-ing

ideal

1M

of M is defined to be

During

the last 15 _years several

_examples

of

_{annihilating polynomials}

of

quadratic

forms have

_appeared

in the literature. Let us first recall some

of these results and

_present

them in the context of

_annihilating

ideals. D. Lewis

_[11]

_gives

an

_{annihilating polynomial}

for

_quadratic

forms of

di-mension n.

Theorem 1

_(Lewis).

Let

_Qn

be the class

_of

all

_{quadratic forms of}

As

_usual,

_(a)

= aR denotes the

principal

ideal

generated by

the element

a

Example

1 e Let

1°~

be the class of all n-fold Pfister forms. Then

Ipn

=

(X2 -

2-X).

Now let us recall the definition of the trace form. Let A be a finite

dimensional étale

_{K -algebra.}

With it we associate the

quadratic

form

which is called the trace

_{form of}

_A/K.

From some

unpublished

result of

P. E. Conner

_[2]

we

_get

the

following

theorem.

Theorem 2

_(Conner).

Let

Tn

be the class

_of

trace

_forms

_of

all

_separable

field

extensions

_{of degree}

n

of

fields

_of

characteristic 54

2.

In a certain _sense, the result of P.E. Conner has been

improved

first

by

P. Beaulieu and T.

_Palfrey

_[1]

and later on

by

D. Lewis and S.

McGarraghy

[12].

Consider a

_separable

field extension

L/K

of

degree n

and let

N/ K

be

a normal closure of

L/K.

Let

f (X)

E

_K[X]

be the minimal

_polynomial

of

a

_primitive

element a of

L/K.

Then Beaulieu and

Palfrey

introduced the

notion of the Galois number of a

_polynomial

_{f (X).}

This number is defined

to be the smallest number _gf such that

_{any gf}

roots of

_{f (X)}

_generate

a

splitting

field of

_{f (X).}

In a _group theoretical context the Galois number of a G-set is defined to be _{the smallest natural number g such that any} group element (j E G

_{with 9}

fixed

_points

acts as the

_identity.

This number

is also called the minimal

_degree

of a

_permutation

representation.

See

[9]

for the determination of Galois numbers of

_doubly

transitive groups. Now

the

_polynomial

of Beaulieu and

_Palfrey

is defined to be

Theorem 3

_{(Beaulieu-Palfrey).}

The

_polynomials

has the

property

that

_{B f(~)}

= 0 _E

W (K),

where 0

is isometric to the trace

form of

the

field

extensions

_given

_by

the

_separable

and irreducible

_polynomials

_{f (X)}

E

_K[X].

Denote the Galois _{group of}

_N/K

_by

_G(N/K).

Then the action of

G(N/K)

on the left cosets of

_{G(NIK)IG(NIL)}

defines a G-set. For _any

G-set S of

cardinality

and _any

_subgroup

U G let

_invu(S)

:=

#{s

E

S

_I

sa = _s for all Q E

_U}

be the number of fixed

_points

of the restricted action. Set

_inv(S)

:=

I

U

G}.

The definition of the

following

any S set

Now set

Theorem 4

_{(Lewis-McGarraghy).}

Let

_L/K

be a

finite

and

separable

field

extension and let

_PG,s(X)

be

_defined

as above. Then

PG,s(X)

annihilates

the trace

_form

_{of L/K.}

Note,

that the result above can be

_generalized

to trace forms of étale

algebras. Using

Springer’s

theorem on the

_lifting

of

_quadratic

forms

ac-cording

to odd

_degree

_extensions,

we

_get

_{annihilating polynomials}

of lower

degree

(see

[8,

Theorem

_2.10]).

Now we come to the definition of the class of

_quadratic

forms we like to

discuss in this _paper.

Definition 2. Let G be a finite _{group and} let H G be a

_subgroup

with

n,EGo,Hu-1

= 1. Then the class

M(G, H)

consists of those

quadratic

forms 0

such that

(1)

there exists an irreducible and

separable polynomial

f (X)

E

_K[X]

with Galois _group

_Gal(f)

_isomorphic

to

_G;

(2)

the action of

_Gal(f)

on the roots of

_{f (X)}

and the action of G on the

left cosets

_G/H

are

equivalent;

(3) ~

and the trace form

_{(K[X]/(j(X)))/ K}

&#x3E; are isometric.

Note,

that the condition on H

_guarantees,

that G acts

_faithfully

on

G/H.

The work of Lewis

_[11]

and Conner

_[2]

_give

_{annihilating polynomials}

for

_{quadratic forms,}

resp. trace forms of dimension n. To finish the

de-termination of the

_{annihilating ideal,}

we have to consider

_signatures.

The

zeros of _resp.

Cn(X)

are

_exactly

those

_integers,

which occur as

signature

values of

_quadratic

forms in _resp.

T~,.

Definition 3. Let M be a class of

quadratic

forms. Then the set of

signa-tures of M is denoted

If

_sign(M)

is

_finite,

the

_{signature polynomial}

of M is

_{given by}

Proposition

1. Let M be a class

of quadratic forms. If

sign(M)

is a

finite

set,

then

Othermise,

we

_{get 1M}

=

(0).

The

_signature

of a trace form

_L/K

&#x3E;

_equals

the number of real

em-beddings

of L into

_JR,

which is the number of real roots of a

_polynomial

f (X)

with L -

_K[X]/(f(X))

_(see

_[13]).

This observation

_gives

rise to the

definition of a

_signature

in a _group theoretical

setting.

Definition 4. Let S be a finite G-set and Q E G with

Q2

= 1. Then

is called the

_signature

of ,S

_according

to a.

is the set of

_signatures

of S.

Observe,

that = From

proposition

3 in

[6]

_we

con-clude

Proposition

2. Let G be a

finite

_group and let H be a

_subgroups

of G

with

n(1EGaHa-1

= 1. Then

2. Burnside

_rings

The

_proofs

of

Conner,

Beaulieu-Palfrey

and

_{Lewis-McGarraghy}

are based

on certain identities in the Burnside

_ring

B(G)

of G and translated into

identities in the Witt

_{ring by}

_applying

a

homomorphism given by

Dress

[5]

(see

also

_[7,

_proposition

_3]).

Let

_B(G)

denote the Burnside

_ring

of G-sets

_(for

more details see

[4,

chapter

11

_§80]).

Let

_XH

denote the G-set

_{given by}

the action of G on the set of

left cosets

_G/H.

Let S =

S(G)

be _a full _set of

nonconjugate subgroups

of

G.

By corollary

80.6 in

_[4]

For _{any a} E

G,O’2

=

_1,

the definition of

signatures given

in definition 4

gives

rise to a

signature homomorphism

Let

_L(G)

:= be the kernel of the total

_signature

ho-momorphism.

For any Galois extension

_N/K

with Galois _group

_G(N/K)

isomorphic

to G there is a

_{ring homomorphism}

_{hN~K :}

G -

W(K).

Let

T(G)

:=

nker(hN/K)

denote the trace ideal of G

_(see

_[6],[7],[8]

for more

~

2 with Galois _group

_{G(N/K) -}

G. Then

_T(G)

C

_L(G).

Theorem 16 in

[7]

states

Theorem 5. Let G be a

finite

_group. Then

L(G)/T(G)

is ca

_{finite 2-group.}

and the

_only

torsions in

_B(G)/T(G)

are 2-torsions.

Together

with

_proposition

1 we conclude

Corollary

1. Let G be a

finite

_groups and let H be a

subgroup

with

Then there is an

_integer

l E

_No

such that

We can choose

21

to be the

_exponent

of

the

finite

abelian.

2-group

L(G)/T(G).

Let

_{(a :}

_b)

:=

{x

E R ~

_xb

C

_a~

be the ideal

_quotient

of the ideals a, b in

the

_ring

R. Then

Since

_IM(G,H)

contains monic

_polynomials,

we

_get

Corollary

2.

_{IM(G,H) is a}

_principal

ideals

_if

and

_{only if}

We introduce some more notation. For any

subgroup

H of G let

res~ :

B(G)

--1-

B(H)

denote the restriction

_homomorphism

_(see

_~7~).

Let

a1,... , an &#x3E; denote the

diagonal

matrix with

_diagonal

entries _{a1,... , an.} For n E N and a matrix A denote the n-fold

_orthogonal

sum

by n

x A :=

n /i

n

Proposition

3. Let

_e(G)

denote the

_exponent

_{of L(G)/T(G).}

_{If any}

sub-group

of

a

2-Sylow

_{subgroup G2 of}

G is a normal

_subgroup

in

G2,

then

Proof.

The

_proof

of

_proposition

4.3 in

_[8]

_implies

_resG2

E

2

_B(G).

Since E

_L(G)

we are done

_{by corollary}

1. 0

The

_following

theorem

_gives

an affirmative answer to a

_question

asked

in

_[7].

With it we are able to translate certain

_problems

on

_annihilating

Theorem 6. Let G be a

finite

_groups with

2-Sylow

subgroup G2.

Then

for

any E

_B(G)

we

_get

Hence the restriction

_homomorphism

induces an

injection

Proof.

From lemma 4.2a in

_[6]

we know E

_T(G2)

_{implies X}

E

_T(G).

Let n :=

ord(G)

and let

N/K

be a Galois extension with Galois _group

G(N/K) -

G2.

Set L :=

K(X1,...,Xn),

where

X1,...,Xn

are

alge-braically independent

indeterminates. The

_{regular representation}

of G de-fines a

monomorphism

G ~

C~({Xl, ... , X~~),

where

C~(~Xl, ... , Xnl)

de-notes the

_symmetric

_{group of} the set

_{~1,...,

Hence G is a

_subgroup

of _{the group} of

_{automorphisms}

_AutK(L).

Set F :=

LG

and

_FZ

:=

LG2.

Since G acts

_transitively

on

{Xl, ... , Xn},

the

_polynomial

is irreducible over F. Hence

Xl

is a

_primitive

element of

_L/F

and of

_L/F2.

Assume,

that

_{Xl, ... , X,,}

m :=

ord(G2)

are the

_conjugates

of

Xl

over

F2.

For any H

G2

we can choose a

_primitive

element aH of

LH / F2

of the

form

_9iXf

_{with 9i}

e

_{L[X1, ...}

n

_F2

=: R.

Let

_{X E}

_T(G).

Then

_{HLIF(X) = 0}

_implies

where H runs over a full set S of

_{nonconjugate subgroups}

of

G2.

For any H E ~S calculate a matrix

MH

of with

_respect

to the

F2-basis

l, aH, ... ,

Hence

_MH

E

_{Gl(nH, R)}

with _nH :_

(GZ : HI

-Since

_hL/F2

_{(res G}

_(x))

= 0

E W (F2),

there is _a matrix A _E

Gl (t, R)

and _a

non-zero

_{polynomial g}

E R with

Let a E N be a

_primitive

element of and let a1 := _{a, a2, ... , am} be

the

_{conj ugates}

of a over

NG2 .

Label these elements

_according

to the action

of

Now we choose

_{algebraically independent}

indeterminates

Y1,..., Ym

and

set

_{Zl .}

:=

_Yl

+~i~2+~~3+...+~"~,...~m

:=

Y1 + amY2 +

+ ... +

By looking

at the Vandermonde determinant we

see that

Zi,...,

Zm

are

algebraically

independent. Replace Xi , ... ,

Xm

by Zi,...,Zyn.

Denote the new

polynomials

_resp. matrices

by

_g,

resp.

MH.

Hence A -

AT - g2 -

_(t/2x

1,

--1

_&#x3E;),

and

_~0.

Since the set of

primitive

elements of a

separable

field extension is a

non-empty

Zariski-open

subset,

there is an

n-tuple

a =

(al, ... , an)

_E

Kn,

such

that

_{g(a1’... ,}

0 and for

_{any H}

E S the element

_9i(a)af

is a

primitive

element of We

_get

3.

_Groups

with

_{quaternion 2-Sylow subgroups}

This section contains a class of

examples,

where

IM(G,H)

is not a

_principal

ideal.

Proposition

4. Let G be a

finite

_group with

2-Sylow subgroup

a

_quaternion

group

of

order 8. Let H G be a

_{subgroup of}

G with = 1.

Then

(a)

ord(H) *

1 mod

2,

(b)

ord(H) -

2 mod

_4,

(c) ord(H) z

0 mod 4 and the

_permutation

_{representation}

_of

G on

G/H

contains

_only

even

permutations.

(2)

Let

_{ord(H) -}

0 mod 4 and _suppose, that the

_permutation

representa-tion

_of

G on

G/H

contains an odd

_permutation.

(a)

Then G is ca serraidirect

product of

G2

and a normal

_{subgroup A}

of

odd order. The

_{conjugation of G2}

on A induces a

monomor-phism (D : G2 Y

Aut(A).

(b)

Let n

:_ ~G : H]

and s :=

_signaXH

with Q E

G2

the

_unique

involution

_{of G2.}

Then

Proof. 1)

see

_proposition

5.2 in

[8].

2(a)

follows from lemma 5.2 in

[8].

2(b):

We use the notation of

§5

in

[8].

By proposition

_3,

theorem 6 and

[6,

proposition

7]

we

_get

2(X - n)(X -

s)

E

_{IMG, H .}

Assertion.

_{X(X - n) (X -}

_s)

E

_IM(G,H)

if

_{8 f}

n.

By

proposition

7 in

_[6]

we have to determine the coefficients mi of XH~ in

Since a = 0 we

_get

_mis

= a’bi

+

2bi (2bi

-n -

s).

_Observe,

that a’ = _{ns -} 0 mod 4 and _{n + s - 2a *} 0 mod 4. Hence

mi == 0 mod 4. We conclude

(2,X)

C

_{(IM(G,H) :}

_{(SignM~G,H)~X)))~}

By

the

preceeding

theorem and

_{by proposition}

5.5 in

_[8]

_IM(G,H)~

Since

_{(2, X)}

is a maximal ideal in

Z[X],

we are done.

The case 8

_n

is left to the reader. D

The smallest

_example

is as follows. The

automorphism

_group of A =

Set G := A x

Q8

and let H be a

_subgroup

of order 4 in G. Then

_{IM(G,H) =}

~x, 2~ . ~~x - 18)(X -

2)) (see

[8,

Example

5.6]).

4. Some more

_2-groups

Lemma 1. Let G be a

_{finite 2-group}

and let H = T &#x3E; be a

subgroups of

order 2 in G with = 1. Then

Here

_CG(r)

denotes the centralizer of T in G.

Proof.

Set n :=

[G : H]

=

ord(G)/2.

Since _T is not contained in the

non-trivial center of

_G,

we

_get

X ~

_by

lemma

3.3(2)

in

[8].

Now

proposition

10 and

_corollary

11 in

_[7]

_gives

the result on and

the

_signature

value.

- -=- _.

’

We

_easily

calculate Hence

5.

_Examples

Finally,

let us summarize some

_examples,

where

_IM(G,H)

is a

principal

ideal.

Theorem 7. In the

_following

cases we

_get

(1)

G has odd order. Then

_{IM(G,H) =}

_{(X - n).}

(2)

G2 is elementary

abelian or

_cyclic.

(3)

G2 is

a dihedral _group

_of

order 2"° &#x3E; 8.

(4)

H = 1.

(5)

G is abelian. ’

(6)

G is a Frobenius _group.

(7)

G is a Zassenhaus

PML(2, q).

(8)

G =

2G2 (q),

q =

32m+ 1 ,

m &#x3E; 1 the Ree _{group in} its

doubly

transi-tive

_{permutation representation}

_{of degree}

_{q3 +}

1 and H a one

_point

(9)

G a _group

of

order 31.

( 1~)

There exists four groups of order &#x3E;

_8,

which contain an element of

order

21.

_{Beside the dihedral group}

_D2l

there are the

_generalized

quaternion

group

Q21,

the

quasidihedral

group

_QD2l

and the group

M(21)

(see ~10~(I.

Satz

_14.9]).

Proof.

For

₍₁₎

see

[3]

_{corollary 1.6.5,}

_resp.

_proposition

17 in

_[7].

Use

proposition

3 and

_[6,

_propositions

5 and

_6]

to _prove

_(2).

(3)

follows from

_proposition

5.1 in

_[8].

4)

If H =

1,

then

BG,H (X)

= X -

n, resp. =

X (X - n).

5)

The condition on H

implies

H = 1.

(6), (7)

and

₍₈₎

follow from

_[8,

_propositions

_6.1,

6.3 and

_3.4].

9)

By

(1), (2),

(3)

and

₍₅₎

it remains to _{consider non-abelian groups} of

order n =

8,16,24

and the _{case n} =

24,

Z/2Z

x

_Z/4Z.

n = 8.

Apply

(3),

resp.

(5)

in the case of the

_quaternion

_group.

n = 16.

Any

subgroup H

G of order &#x3E; 4 has _a non-trivial intersection

with the center of G

_{(see [14]).}

Now use lemma 1.

n = 24. The result for

G2 -

Z/2Z

x

_Z/4Z

follows from some

_unpublished

determination of the

exponent

of

_L(G)/T(G)

for G =

Z/2Z

x

Z/2IZ.

Since there is no

_injection

Q8 o

Aut(Z/3Z),

we are done

_{by proposition}

4.

(10)

follows from lemma

_1,

since _any

_subgroup

H of G of order &#x3E; 4 has a

non-trivial intersection with the center of G. 0

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Martin EPKENHANS Fb Mathematik Universitat Paderborn

D-33095 Paderborn Deutschland