On prime factors of class number of cyclotomic fields

0.23 MB·English

by  Roland Queme

7 On prime factors of class number of cyclotomic fields
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Roland Quˆeme
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2006 jan 09

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Contents
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1 Some definitions 2
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2 On Kummer and Stickelberger relation 3
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5 3 Polynomial congruences mod p connected to the p-class group C 6
p
v
3.1 Stickelberger relation for prime ideals q of inertial degree f = 1 . . . . . . 6
3
2 3.2 Stickelberger relation for prime ideals q of inertial degree f > 1 . . . . . . 7
7 3.3 Polynomial congruences mod p2 connected to the p2-class group C . . . 9
9 p2
0
6 4 On prime factors h = p of the class number of the p-cyclotomic field 11
0 6
4.1 The general case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
/
h 4.2 The case of complex quadratic fields contained in K . . . . . . . . . . . . . 15
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a 4.3 The case of biquadratic fields contained in K . . . . . . . . . . . . . . . . 17
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v 5 A numerical MAPLE algorithm 18
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Abstract
r
a
Let p be an odd prime. Let K = Q(ζ) be the p-cyclotomic number field. Let
v be a primitive root mod p and σ : ζ ζv be a Q-isomorphism of the extension
→
K/Q generating the Galois groupG of K/Q. Following theconventions of Ribenboim
in [8], for n Z, the notation v is understood by v = vn mod p with 1 v
n n n
p 1. Let P∈(X) = p−2Xiv Z[X] be the Stickelberger polynomial≤. P(σ≤)
− Pi=0 −i ∈
annihilates the class group C of K. There exists a polynomial Q(X) Z[G] such
∈
that P(σ) (σ v) = p Q(σ) and such that Q(σ) annihilates the p-class group C
p
× − ×
of K.
1
These result allow:
1. to describe the structure of the relative class group C−,
2. to give some explicit congruences in Z[v] mod p for the p-class group of K (the
subgroup of exponent p of C),
3. to give some explicit congruences in Z[v] mod h for the h-class group of K for
the prime divisors h = p of the class number h(K).
6
4. We detail attheendthecase of class numberof quadraticandbiquadraticfields
contained in the cyclotomic field K.
5. As application, we give a MAPLE algorithm which describes the structure of
the relative class group C− of the cyclotomic field K for all the prime numbers
p < 500.
This article is at elementary level.
Remark: we have not found in the literature some formulations corresponding to
theorems 3.2 p. 7, 4.1 p. 12, 4.3 p. 13, 4.6 p. 16, 4.7 p. 17, and 4.8 p. 17 and to
MAPLE algorithm described in section 5 p 18.
1 Some definitions
In this section we give some definitions and notations on cyclotomic fields and p-class
group used in this paper.
1. Let p be an odd prime. Let F be the finite field of cardinal p and F∗ its
p p
multiplicative group. Let ζ bearootof thepolynomialequation Xp−1+Xp−2+
+X +1 = 0. Let K be the p-cyclotomic field K = Q(ζ) and O its ring of
K
···
integers. LetK+ bethemaximal totally realsubfieldofK. Letv beaprimitive
root mod p. In this paper, following Ribenboim conventions in Ribenboim [8]
for any n Z, we note v = vn mod p with 1 v p 1. Let G be the
n n
∈ ≤ ≤ −
Galois group of the extension K/Q. Let σ : ζ ζv be a Q-isomorphism of the
→
extension K/Q generating G. Let λ =ζ 1. The prime ideal of K lying over p
−
is π = λO .
K
2. Let C be the class group of K. Let C be the p-class of K (the subgroup of
p
exponent p of C ). Let C+ be the p-class group of K+. Then C = C+ C−
p p p ⊕ p
where C− is called the relative p-class group. Let r− be the rank of C−.
p p
3. C− isthedirectsumofr− subgroupsΓ oforderpannihilatedbyσ µ F [G]
p k − k ∈ p
with µ F∗, µ = v where m is a natural integer m , 1 m p−3,
k ∈ p k 2mk+1 k k ≤ k ≤ 2
(1) C− = r− Γ .
p ⊕k=1 k
2
2 On Kummer and Stickelberger relation
Stickelberger relation was already known by Kummer under the form of Jacobi resol-
vents for the cyclotomic field K, see for instance Ribenboim [8] (2.6) p. 119. In this
section we derive some elementary properties from Stickelberger relation.
1. Let q = pbean oddprime. Letζ bearootof theminimalpolynomialequation
q
6
Xq−1+Xq−2+ +X +1= 0. Let K = Q(ζ ) be the q-cyclotomic field. Let
q q
···
K =Q(ζ ,ζ ). ThenK is the compositum KK . Thering of integers of K
pq p q pq q pq
is O .
Kpq
2. Let q be a prime ideal of O lying over the prime q. Let f be the order of
K
q mod p and m = N (q) = qf. If ψ(α) = a is the image of α O under
Kp/Q ∈ K
the natural map ψ : O O /q, then for ψ(α) = a 0 define a character
K K
→ 6≡
(p)
χ on F = O /q by
q m K
α −1 α
(p)
(2) χ (a) = = ,
q
{q} {q}
p p
where α = ζc for some natural integer c, is the pth power residue character
{q}
mod q. We define the Gauss sum
(3) g(q) = X (χq(p)(x)×ζqTrFm/Fq(x)) ∈ OKpq.
x∈Fm
It follows that g(q) O . Moreover g(q)p O , see for instance Mollin [6]
∈ Kpq ∈ K
prop. 5.88 (c) p. 308.
3. The Stickelberger relation is classically:
(4) g(q)pO = qS,
K
with S = p−1t ̟−1, where ̟ Gal(K/Q) is given by ̟ :ζ ζt (see for
Pt=1 × t t ∈ t →
instance Mollin [6] thm. 5.109 p. 315).
ThefourfollowinglemmasarederivedinanelementarywayfromtheStickelberger
relation.
Lemma 2.1. If q 1 mod p then the Gauss sum g(q) Z[ζ].
6≡ ∈
Proof.
1. Letubeaprimitiveroot modq. Letτ : ζ ζu beaQ-isomorphismgenerating
q → q
Gal(K /Q). The Q-isomorphism τ is extended to a K -isomorphism of K by
q p pq
τ :ζ ζu, ζ ζ . Then g(q)p Z[ζ ] and so
q → q p → p ∈ p
τ(g(q))p =g(q)p,
3
and it follows that there exists a natural integer ρ with ρ < p such that
τ(g(q)) = ζρ g(q).
p ×
Then N (τ(g(q))) = ζ(q−1)ρ N (g(q)) and so ζρ(q−1) = 1.
Kpq/K × Kpq/K
2. If q 1 mod p, it implies that ζρ = 1 and so that τ(g(q)) = g(q) and thus that
6≡
g(q) O .
K
∈
Lemma 2.2. Let S = p−1̟−1 t where ̟ is the Q-isomorphism of the extension
Pt=1 t × t
K/Q given by ̟ :ζ ζt of K. Let P(σ) = p−2σi v Z[G]. Then S = P(σ).
t → Pi=0 × −i ∈
Proof. Let us consider one term ̟−1 t. Then v = v is a primitive root mod p
t × −1 p−2
and so there exists one and one i such that t = v−i. Then ̟v−i : ζ → ζv−i and so
̟−1 : ζ ζvi and so ̟−1 = σi (observe that σp−1 v = 1), which achieves
v−i → v−i × −(p−1)
the proof.
Lemma 2.3.
p−2 p−2
(5) P(σ) = Xσi v−i = v−(p−2) Y (σ vk) +p R(σ),
× ×{ − } ×
i=0 k=0, k6=1
where R(σ) Z[G] with deg(R(σ)) < p 2.
∈ −
p−2
Proof. Let us consider the polynomial R (σ) = P(σ) v (σ v )
0 − −(p−2)×{Qk=0, k6=1 − k }
in F [G]. Then R (σ) is of degree smaller than p 2 and the two polynomials
p 0
−
p−2σiv and p−2 (σ v ) take a null value in F [G] when σ takes the p 2
Pi=0 −i Qk=0, k6=1 − k p −
different values σ = v for k = 0,...,p 2, k = 1. Then R (σ) = 0 in F [G] which
k 0 p
− 6
leads to the result in Z[G].
Lemma 2.4.
(6) P(σ) (σ v) = p Q(σ),
× − ×
4
where Q(σ) = p−2δ σi Z[G] is given by
Pi=1 i× ∈
v v v
−(p−3) −(p−2)
δ = − ,
p−2
p
v v v
−(p−4) −(p−3)
δ = − ,
p−3
p
.
.
.
(7)
v v v
−(i−1) −i
δ = − ,
i
p
.
.
.
1 v v
−1
δ = − ,
1
p
with p < δ 0.
i
− ≤
Proof. We start of the relation in Z[G]
p−2
P(σ) (σ v) = v−(p−2) Y(σ vk)+p R(σ) (σ v)= p Q(σ),
× − × − × × − ×
k=0
withQ(σ) Z[G]because p−2(σ v ) = 0inF [G]andso p−2(σ v ) = p R (σ)
∈ Qk=0 − k p Qk=0 − k × 1
in Z[G]. Then we identify in Z[G] the coefficients in the relation
(v σp−2+v σp−3+ +v σ+1) (σ v) =
−(p−2) −(p−3) −1
··· × −
p (δ σp−2+δ σp−3+ +δ σ+δ ),
p−2 p−3 1 0
× ···
where σp−1 = 1.
Remarks:
1. Observe that we have more generally for the indeterminate X the algebraic
identity in Z[X]
(8) P(X)(X v) = p Q(X)+v(Xp−1 1).
− × −
2. Observe that, with our notations, δ Z, i = 1,...,p 2, but generally
i
∈ −
δ 0 mod p.
i
6≡
3. We see also that p < δ 0. Observe also that δ = v−(p−2)−v =0.
− i ≤ 0 p
5
3 Polynomial congruences mod p connected to
the p-class group C
p
We give someexplicit polynomial congruences inZ[v]modpconnected to therelative
p-class group C− of O . We apply successively the Stickelberger relation to prime
p K
ideals q of inertial degree f = 1 and of inertial degree f > 1. We recall that r− is the
p-rank of the relative p-class group C− of K.
p
3.1 Stickelberger relation for prime ideals q of inertial
degree f = 1
Theorem 3.1. r− (σ v ) divides Q(σ) in F [G] and
Qk=1 − 2mk+1 p
p−2
v v v
−(i−1) −i
(9) Q(v2mk+1) = Xv(2mk+1)×i×( −p × ) ≡ 0 mod p,
i=1
where m is defined in relation (1) p. 2.
k
Proof. From Kummer, the group of ideal classes of K is generated by the classes of
prime ideals of degree 1 (see for instance Ribenboim [8] (3A) p. 119). Let q a prime
ideal of inertial degree 1 whose class Cl(q) C− is annihilated by σ µ with µ =
∈ p − k k
v . We start of g(q)pO = qP(σ) and so g(q)p(σ−v)O = qP(σ)(σ−v) = qpQ(σ)
2mk+1 K K
and thus g(q)(σ−v)O = qQ(σ). It can be shown that g(q)σ−v K, see for instance
K
∈
Ribenboim [9] F. p. 440. Therefore Q(σ) annihilates the ideal class Cl(q) and the
congruence follows.
Remarks:
1. Observe that δ can also be written in the form δ = [v−i×v] where [x] is the
i i − p
integer part of x, similar form also known in the literature.
2. Observe that it is possible to get other polynomials of Z[G] annihilating the
relative p-class group C−: for instance from Kummer’s formula on Jacobi cy-
p
clotomic functions we induce other polynomials Q (σ) annihilating the relative
d
p-class group C− of K : If 1 d p 2 define the set
p ≤ ≤ −
I = i 0 i p 2, v +v > p
d { | ≤ ≤ − (p−1)/2−i (p−1)/2−i+indv(d) }
where the index ind (d) is the minimal integer s such that d = v . Then the
v s
polynomials Q (σ) = σi for d = 1,...,p 2 annihilate the p-class C of
d Pi∈Id − p
K, see for instance Ribenboim [8] relations (2.4) and (2.5) p. 119.
6
3. See also in a more general context Washington, [11] corollary 10.15 p. 198.
4. It is easy to verify the consistency of relation (9) with the table of irregular
primes and Bernoulli numbers in Washington, [11] p. 410.
5. See section 5 p. 18 with an algorithm improving this result.
3.2 Stickelberger relation for prime ideals q of inertial
degree f > 1
Let q be a prime ideal of O with Cl(q) C . In this section we apply Stickelberger
K p
∈
relation to the prime ideals q of inertial degree f > 1 with the method used for the
primeideals of inertial degree 1 in section 3 p. 6. Observe, from lemma 2.1 p. 3, that
f > 1 implies that g(q) O , property used in this section (by opposite g(q) K
K
∈ 6∈
when f = 1).
A definition: we say that the prime ideal c of a number field M is p-principal if
the component of the class group < Cl(c) > in p-class group D of M is trivial.
p
Theorem 3.2. Let q be an odd prime with q = p. Let f be the order of q mod p and
m = p−1. Let q be a prime ideal of O lying6over q with Cl(q) C−. If f > 1 then
f K ∈ p
1. g(q) OK and g(q)OK = qP1(σ) where
∈
m−1 f−1v
(10) P1(σ) = X(Pj=0 −(i+jm)) σi Z[G].
p × ∈
i=0
2. There exists a natural integer n with 1 n p 2 such that σ v divides
n
≤ ≤ − −
P (σ) in F [G] and σ v annihilates Cl(q).
1 p n
−
Proof.
1. N (q) = qf and q = qσm = = qσ(f−1)m. From Stickelberger relation
K/Q ···
g(q)pZ[ζ ] = qP(σ) where P(σ) = m−1 f−1σi+jmv . Observe that,
p Pi=0 Pj=0 −(i+jm)
from hypothesis, q = qσm = = qσ(f−1)m so Stickelberger’s relation implies
···
that g(q)pO = qP(σ) with
K
m−1f−1 m−1 f−1v
P(σ) = X Xσiv−(i+jm) = p X(Pj=0 −(i+jm)) σi,
× p ×
i=0 j=0 i=0
f−1
where ( v )/p Z because v 1 0 mod p.
Pi=0 −(i+jm) ∈ −m− 6≡
7
P(σ)
2. Let P (σ) = . Then from below P (σ) Z[G]. From lemma 2.1 p. 3 we
1 p 1 ∈
know that f > 1 implies that g(q) O . Therefore
K
∈
g(q)pO = qpP1(σ), g(q) O ,
K K
∈
and so
g(q)O = qP1(σ), g(q) O .
K K
∈
Then P (σ) annihilates Cl(q) and thus there exists 1 < n p 2 such that
1
≤ −
(σ v ) P (σ).
n 1
− |
Remarks
1. For f = 2 the value of polynomial P (σ) obtained from this lemma is P (σ) =
1 1
(p−3)/2σi.
Pi=1
2. Let q be a prime not principal ideal of inertial degree f > 1 with Cl(q) C .
p
∈
The two polynomials of Z[G ], Q(σ) = p−2(v−(i−1)−v−iv) σi (see thm 3.1)
p Pi=0 p ×
andP (σ) = m−1(Pjf=−01v−(i+jm)) σi (seelemma3.2)annihilatetheidealclass
1 Pi=0 p ×
Cl(q). When f > 1 the lemma 3.2 p. 7 supplement the theorem 3.1 p. 6.
3. This result explains that, when f increases, the proportion of p-principal ideals
q increases.
It is possible to derive some explicit congruences in Z from this theorem.
Corollary 3.3. Let q be an odd prime with q = p. Let f be the order of q mod p and
let m = p−1. Let q be an prime ideal of O ly6 ing over q. Suppose that f > 1.
f K
1. If the ideal q is non p-principal there exists a natural integer l, 1 l < m such
≤
that
m−1 f−1v
(11) X(Pj=0 −(i+jm)) vlfi 0 mod p,
p × ≡
i=0
2. If for all natural integers l such that 1 l < m
≤
m−1 f−1v
(12) X(Pj=0 −(i+jm)) vlfi 0 mod p,
p × 6≡
i=0
then q is p-principal
Proof.
8
1. Suppose that q is not p-principal. Observe at first that congruence (11) with
l = m should imply that m−1( f−1v )/p) 0 mod p or
Pi=0 Pj=0 −(i+jm) ≡
m−1f−1
X Xv−(i+jm) 0 mod p2,
≡
i=0 j=0
whichisnotpossiblebecausev−(i+jm) = v−(i′+j′m) impliesthatj = j′ andi= i′
and so that m−1 f−1v = p(p−1).
Pi=0 Pj=0 −(i+jm) 2
2. The polynomial P (σ) of lemma 3.2 annihilates the non p-principal ideal q in
1
F [G] only if there exists σ v dividing P (σ) in F [G]. From qσm−1 = 1 it
p n 1 p
−
follows also that σ v σm 1. But σ v σm v and so σ v v 1,
n n nm n nm
− | − − | − − | −
thus nm 0 mod p 1, so n 0 mod f and n = lf for some l. Therefore if q
≡ − ≡
is non p-principal there exists a natural integer l, 1 l < m such that
≤
m−1 f−1v
(13) X(Pj=0 −(i+jm)) vlfi 0 mod p,
p × ≡
i=0
3. The relation (12) is an immediate consequence of previous part of the proof.
3.3 Polynomial congruences mod p2 connected to the
p2-class group C
p2
Let C− be the subgroup of exponent p2 (so with elements of order dividing p2) of
p2
the relative class group C− of K.
1. We have seen in relation (1) p. 2 that the relative p-class group C− can be seen
p
as a direct sum C− = r− Γ where Γ is a cyclic subgroup of C− annihilated
p ⊕i=1 i i p
by σ µ with 2 µ p 2.
i i
− ≤ ≤ −
2. C− can be seen as a direct sum
p2
(14) C− = r− ∆ ,
p2 ⊕i=1 i
where ∆ is a cyclic group with Γ ∆ and whose order divides p2 .
i i i
⊂
3. Suppose that ∆ is of order p2. Show that ∆ is annihilated by σ (µ +a p)
i i i i
−
with a natural integer 1 a p 1:
i i
≤ ≤ −
(a) From Kummer, there exist some prime ideals Q of O with Cl(Q ) ∆ ,
i K i i
∈
Cl(Qp) Γ and < Qp(σ−µi) > principal as seen in previous sections.
Thereifor∈e <iQσ−µi > isiof order p.
i
9
(b) Cl(Qµi) ∆ . In the other hand Cl(Qσ) ∆ : if not Cl(Qσ) should
have ait lea∈st onie component c ∆ , j =ii an∈d soi Cl(Qpσ) shouild have a
componentcp ∆p = Γ , j =ji,∈conjtradi6ction. ThereforeiCl(Qσ−µi) ∆ .
j ∈ j j 6 i ∈ i
Then, from Qp(σ−µi) principal, it follows that Cl(Qσ−µi) Γ because ∆
i i ∈ i i
is cyclic of order p2.
(c) Thus there exists a , 1 a p 1, such that Cl(Qσ−µi)= Cl(Qaip) and
so Qσ−µi−ai is prinicipal≤. i ≤ − i i
i
In this section we examine the case of subgroups ∆ of order p2. Let us note ∆ for
i
one of this groups annihilated by σ (µ+ap), a = 0.
− 6
Theorem 3.4. µ verifies the two congruences
p−2
v v v
Xµiδi 0 mod p with δi = −(i−1)− −i ,
≡ p
i=0
(15)
p−2 p−2
Xµp−2+iδi+(µp−1 1)Xiµi−1δi 0 mod p2.
− ≡
i=0 i=1
Proof.
1. Thereexists primeideals Qof O withCl(Q) ∆, henceQp2 principalandQp
K
∈
not principal. From Stickelberger relation g(Q)pO = QP(σ) where P(σ) has
K
been defined in lemma 2.2 p. 4. Then g(Q)p(σ−v)O = QP(σ)(σ−v), hence from
K
lemma 2.4 p. 4 we get g(Q)p(σ−v)O = QpQ(σ), hence g(Q)(σ−v)O = QQ(σ).
K K
Weknow,forinstancefromRibenboim[9]F.p. 440thatg(Q)σ−v K soQQ(σ)
∈
is principal. But Qσ−(µ+ap) is principal hence QQ(µ+ap) is principal, and thus
(16) Q(µ+ap) 0 mod p2.
≡
2. From lemma 2.4 p. 4, Q(σ) = p−2σiδ where δ = v−(i−1)−v−iv. From relation
Pi=0 i i p
(16)
p−2
X(µ+ap)iδi 0 mod p2,
≡
i=0
hence
p−2 p−1
Xµiδi+apXiµi−1δi 0 mod p2.
≡
i=0 i=1
From theorem 3.1 p. 6 applied to the ideal Qp Γ of order p,
∈
p−2
Xµiδi 0 mod p.
≡
i=0
10