Journal de Théorie des Nombres de Bordeaux

Helmut Maier

Michael Th. Rassias

Prime Avoidance Property of $k$-th Powers of Piatetski–Shapiro Primes

Martin Djukanović

Families of split Jacobians with isogenous components

Kim Klinger-Logan

Kalani Thalagoda

Tian An Wong

A Dedekind–Rademacher cocycle for Bianchi groups

Guillem Tarrach

The $p$-arithmetic homology of mod $p$ representations of $\mathrm{GL}_2(\mathbb{Q}_p)$

Erik Holmes

On the shapes of pure prime-degree number fields

Byoung Du (BD) Kim

Local conditions of adjoint representations with supersingular reduction, and representable functors of deformations with higher weight liftings

François Maucourant

Size of discriminants of periodic geodesics on the modular surface

Lassina Dembélé

René Schoof

GRH and finite flat group schemes over $\mathbf{Z}$

Gabriel Raposo

A Refinement of Pohst’s Inequality

Jonathan Jenvrin

Jonathan JENVRIN
Explicit lower bounds for the height in Galois extensions of number fields
Tome 37, no 2 (2025), p. 469-477.
https://doi.org/10.5802/jtnb.1329
© Les auteurs, 2025.
Cet article est mis à disposition selon les termes de la licence
CREATIVE COMMONS ATTRIBUTION – PAS DE MODIFICATION 4.0 FRANCE.
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Journal de Théorie des Nombres
de Bordeaux 37 (2025), 469–477
Explicit lower bounds for the height in Galois
extensions of number fields
par Jonathan JENVRIN
Résumé. Amoroso et Masser ont prouvé que, pour tout réel ϵ > 0, il existe une
constante c(ϵ) > 0, avec la propriété suivante : pour tout nombre algébrique
α tel que Q(α)/Q est une extension galoisienne, la hauteur de α est soit nulle,
soit au moins c(ϵ)[Q(α) : Q]−ϵ. Dans le présent article, nous établissons une
version explicite de ce théorème.
Abstract. Amoroso and Masser proved that for every real ϵ > 0, there
exists a constant c(ϵ) > 0, with the property that, for every algebraic number
α such that Q(α)/Q is a Galois extension, the height of α is either 0 or at
least c(ϵ)[Q(α) : Q]−ϵ. In the present article, we establish an explicit version
of the aforementioned theorem.
1. Introduction
In this article, we let Q be a fixed algebraic closure of Q. For an algebraic
number α of degree d over Q, we denote by h(α) its absolute logarithmic
Weil height defined as
h(α) = 1
d
(
log
(
|a|
d∏
i=1
max(1, |αi|)
))
where a is the leading coefficient of the minimal polynomial of α over Z,
and α1, . . . , αd are the conjugates of α over Q. Notice in particular that
h(α) ≥ 0 for all α ∈ Q.
While, by Kronecker’s theorem (see for instance [9, Theorem 1.5.9]), it
is well-known that h(α) = 0 if and only if α is either 0 or a root of unity,
in [14] Lehmer raised the question of whether there exists a constant c > 0
such that
h(α) ≥ c
d
whenever h(α) is not zero. The existence of such a constant is nowadays
known as Lehmer’s conjecture and has been proved for various classes of
algebraic numbers, but is still open in general. For instance, the conjecture is
obviously true for α not a unit with c = log(2). While for α non-reciprocal
Manuscrit reçu le 7 février 2024, révisé le 24 octobre 2024, accepté le 16 novembre 2024.
2020 Mathematics Subject Classification. 11G50.
Mots-clefs. Height bounds, Galois extensions, Lehmer’s problem.
470 Jonathan Jenvrin
(which is always the case for d odd) its validity was proved in [19] with
c = 3h(θ) = log(θ), where θ is the real root > 1 of X3 − X − 1. The most
notable progress toward Lehmer’s conjecture is Dobrowolski’s result [11,
Theorem 1], later made explicit by Voutier in [20, Theorem on p. 83],
proving that if h(α) ̸= 0, then
h(α) ≥ 14d
( log log d
log d
)3
.
This implies that for any ϵ > 0, there is an explicit constant c̃(ϵ) > 0 such
that either h(α) = 0 or h(α) ≥ c̃(ϵ)d−1−ϵ.
While Dobrowolski’s result remains the only unconditional one on this
problem, it is possible to prove that specific classes of algebraic numbers
satisfy even stronger variants of Lehmer’s conjecture, such as the Bogo-
molov property introduced by Bombieri and Zannier in [10]. A set of alge-
braic numbers S has the Bogomolov property (B) if there exists a constant
c = c(S) > 0 such that for every α ∈ S, either h(α) = 0 or h(α) ≥ c.
A set of algebraic numbers that has garnered attention in recent years is
that of all α ∈ Q such that Q(α)/Q is Galois.
Amoroso and David proved that Lehmer’s conjecture holds for elements
of this set (see [2, Corollary 1.7]). Later, Amoroso and Masser [6, The-
orem 3.3] showed an even stronger result: for any ϵ > 0, there exists a
positive effective constant c(ϵ) such that, for every α ∈ Q of degree d over
Q and not a root of unity, such that Q(α)/Q is Galois, one has
h(α) ≥ c(ϵ)d−ϵ.(⋆)
This strong result raises the question of whether the set of algebraic
numbers that generate a Galois extension over Q satisfies Property (B). A
natural way to tackle this question is to fix the Galois group G of Q(α)/Q.
The answer is positive when G is abelian [5], dihedral [8, Corollary 1.3], has
an exponent bounded by an absolute constant [3, Corollary 1.7] or has odd
order (since the field of totally real numbers satisfies Property (B) by [18,
Corollary 1]). An even stronger result has been proven for some classes of
generators of Galois extensions of Q of group Sn in [1, Theorem 1.2 and
Theorem 1.3], or An in [12, Theorem 1.4 and Theorem 1.5]; in these cases
the height of such generators goes to infinity with n. All these results give
evidences for a positive answer to the aforementioned question, which is
still open in general.
The goal of our article is to give an explicit version of (⋆). Our main
result is the following:
Lower bounds for the height in Galois extensions 471
Theorem 1.1. Suppose α ∈ Q∗ is of degree d over Q. If α is not a root of
unity and Q(α)/Q is Galois, then
h(α) ≥ 10−8 exp
(
−492 log(3d)
3/4 log(log(3d))
)
.
As an easy corollary, we obtain an explicit version of Amoroso and
Masser’s result [6, Theorem 3.3].
Corollary 1.2. For every ϵ > 0 and for every α of degree d, not a root of
unity, such that Q(α)/Q is Galois, one has
h(α) ≥ c(ϵ)d−ϵ
where
c(ϵ) = 10−8
(1
3
)ϵ
exp
(
−181
(724
5ϵ
)4
−
(724
5ϵ
)5)
.
Theorem 1.1 and Corollary 1.2 are proved in Section 3. Our proof strat-
egy relies on that of Amoroso and Masser’s result in [6, Theorem 3.3].
In particular, we divide our proof in two cases, according to the rel-
ative magnitude of the multiplicative rank ρ(α) of the subgroup of Q×
generated by the conjugates of α, and the quantity log(3 deg(α))1/4, where
d(α) = [Q(α) : Q]. When ρ(α) > log(3 deg(α))1/4, we conclude using a
result of Amoroso and Viada [7], which we recall in Theorem 2.1, bound-
ing from below the products of heights of multiplicatively independent al-
gebraic numbers. On the other hand, when ρ(α) ≤ log(3 deg(α))1/4 we
conclude by applying a result of Amoroso and Delsinne [4], providing an
explicit relative version of Dobrowolski’s lower bound, where the degree
deg(α) = [Q(α) : Q] is replaced by a relative degree [L(α) : L], where
L/Q is a finite abelian extension. This is in contrast with Amoroso and
Masser’s proof of (⋆), which uses an older higher-dimensional generaliza-
tion of Dobrowolski’s lower bound proven by Amoroso and David [2] when
ρ(α) > log(3d(α))1/4, and the relative Dobrowolski lower bound proven by
Amoroso and Zannier when ρ(α) ≤ log(3d(α))1/4.
2. Auxiliary Results
We state two important results that will be the key ingredients in our
proof. The first was proved by Amoroso and Viada in [7, Corollary 1.6],
where, more generally, they give an explicit version of a generalized Do-
browolski result on Lehmer’s problem.
Theorem 2.1 ([7, Corollary 1.6]). Let α1, . . . , αn be multiplicatively inde-
pendent algebraic numbers in a number field K. Then
h(α1) . . . h(αn) ≥ [K : Q]−1(1050n5 log(3[K : Q]))−n
2(n+1)2 .
472 Jonathan Jenvrin
The second result was proved by Amoroso and Delsinne in [4, Théo-
rème 1.3], and provides a relative version of Dobrowolski’s lower bound,
when the base field considered is abelian. We recall a special case of this
theorem, which is enough for our purposes.
Theorem 2.2 ([4, Théorème 1.3]). Let α ∈ Q∗ be not a root of unity
and let L be a number field. Then if L/Q is a finite abelian extension and
D = [L(α) : L], we have
h(α) ≥ D−1 log log(5D)
3
log(2D)4 .
Proof. We apply [4, Théorème 1.3], with L = L and K = Q. Notice that
the lower bound in [4] depends on a certain quantity (g([K : Q]) · ∆K)−c
which is 1 for K = Q. This gives the desired result. □
The following lemma is a relative version of [6, Lemma 2.2].
Lemma 2.3. Let F/K be a finite Galois extension and α ∈ F \ {0}. Let
α1, . . . , αd be the conjugates of α over K, i.e. the orbit of α under the
action of Gal(F/K). Moreover let ρ be the rank of the multiplicative group
generated by α1, . . . , αd, and suppose that ρ ≥ 1. Then there exists a subfield
L ⊂ F which is Galois over K of degree [L : K] ≤ n(ρ), such that F
contains a primitive k-th root of unity ζk and αk ∈ L, where k is the order
of the group of roots of unity in F . We can take
n(ρ) = ρ!2ρ for ρ = 1, 3, 5 and ρ > 10.
Otherwise we have
n(ρ) 2 4 6 7 8 9 10
ρ 12 1152 103680 2903040 696729600 1393459200 8360755200
Proof. Define βi = αki for 1 ≤ i ≤ d and L = K(β1, . . . , βd). We have
L ⊂ F because F/K is Galois, and we easily check that L/K is Galois. The
Z−module
M = {βa11 . . . β
ad
d | a1, . . . , ad ∈ Z}
is torsion free by the choice of k and so, by the classification of finitely
generated abelian groups ([13, Theorem 8.1 and Theorem 8.2]), is free, of
rank ρ. This shows that the action of Gal(L/K) over M defines an injective
representation Gal(L/K) → GLρ(Z). We obtain that Gal(L/K) identifies
to a finite subgroup of GLρ(Z). To conclude, we use a theorem stated by
Feit in 1996, and proved by Rémond in [15, Théorème 7.1], which computes
the largest cardinality of a finite subgroup of GLρ(Z). □
Remark 2.4. The constant n(ρ) in Lemma 2.3 is somehow optimal, since
it is equal to the largest cardinality of a finite subgroup of GLρ(Z).
Lower bounds for the height in Galois extensions 473
We continue this section with the following lemma, which gives an ex-
plicit upper bound for the quotient of an integer over his Euler’s totient.
Lemma 2.5. Let ϕ be the Euler’s totient function. For every positive in-
teger n ≥ 1, we have
n
ϕ(n) ≤
log(log(3n))
log(log(3))
and
ϕ(n) ≥
√
n
2 .
Proof. The first inequality can be easily deduced from [17, Theorem 15] for
n ≥ 100 and verified for smaller values of n. Indeed, if we denote by γ the
Euler’s constant, then for n ≥ 100, we have
eγ log(log(n)) + 2.50637log(log(n)) ≤ 3 log(log(3n)) ≤
log(log(3n))
log(log(3)) .
The second inequality is well-known, and can be deduced easily from the
previous one. □
We conclude this section by recalling Stirling’s upper bound ([16, (1)
and (2) on p. 26]), which will be use in the next section.
Lemma 2.6. For every n ≥ 1, we have
n!
nn
≤
√
2πne
1
12n e−n.
3. Proof of Theorem 1.1
We can now establish the proof of Theorem 1.1. To this end, we fix for
the rest of this section an algebraic number α ∈ Q such that Q(α)/Q is
Galois. Firstly, we will prove a lower bound for h(α) depending on the
multiplicative rank ρ of the conjugates of α, and on the degree d of Q(α)
over Q.
Lemma 3.1. Let
g1(ρ, d) = min1≤r≤ρ
(
d1/r
(
1050r5 log(3d)
)r(r+1)2)
and
g2(ρ, d) = 6.5 · 107ρρ+5 log
(
log
(
6d2
))5
.
Then:
h(α)−1 ≤ min(g1(ρ, d), g2(ρ, d)).
474 Jonathan Jenvrin
Proof. Let α1, . . . , αρ be multiplicatively independent conjugates of α, and
fix r ∈ {1, . . . , ρ}. Since Q(α)/Q is Galois by assumption, we have that
α1, . . . , αr ∈ Q(α). Therefore, applying Theorem 2.1 to α1, . . . , αr we see
that
h(α) = (h(α1) · · · h(αr))1/r ≥ d−1/r
(
1050r5 log(3d)
)−r(r+1)2
,
which implies that h(α) ≥ g1(ρ, d)−1.
Let k be the number of roots of unity in Q(α). Then, by Lemma 2.3
there exists a subfield L ⊂ Q(α) whose degree can be bounded as follows:
[L : Q] ≤ 135ρ!2ρ−1,
such that the extension L/Q is Galois and αk ∈ L. Set M = Q(ζk). We
have M(α) = Q(α) = L(α). Notice that
[M(α) : M ] = [L(α) : L] [L : Q][M : Q] .
Since α is a root of Xk − αk ∈ L[X], we have that [L(α) : L] ≤ k. We
also notice that ϕ(k) = [M : Q]. Therefore, we obtain the following chain
of inequalities:
[M(α) : M ] ≤ k [L : Q][M : Q] ≤
k
ϕ(k) [L : Q].
By Lemma 2.5, we have that
k
ϕ(k) ≤
log(log(3k))
log(log(3)) ,
so we obtain
[M(α) : M ] ≤ 135log(log(3k))log(log(3)) ρ!2
ρ−1.
Since k is the number of roots of unity contained in L, we have that
ϕ(k) ≤ d. By the last statement of Lemma 2.5, we have that
√
k/2 ≤ ϕ(k),
hence k ≤ 2d2 and
[M(α) : M ] ≤ 135log
(
log
(
6d2
))
log(log(3)) ρ!2
ρ−1 ≤ 718ρ!2ρ log
(
log
(
6d2
))
.
By Lemma 2.6, applied to ρ, we have that
[M(α) : M ] ≤ 718ρρ
√
2πρe
1
12ρ e−ρ2ρ log
(
log
(
6d2
))
≤ 1440ρρ log
(
log
(
6d2
))
By Theorem 2.2, applied to Q(ζk)/Q, we have
h(α) ≥ [M(α) : M ]−1 log(log(5[M(α) : M ]))
3
log(2[M(α) : M ])4 .
Lower bounds for the height in Galois extensions 475
Since
x 7−→ 1
x
log(log(5x))3
log(2x)4
is a decreasing map on [1, +∞[, we deduce, by setting
X(ρ, d) = 1440ρρ log
(
log
(
6d2
))
that
h(α)−1 ≤ X(ρ, d) log(2X(ρ, d))4
= 1440ρρ log
(
log
(
6d2
))
log
(
4884ρρ log
(
log
(
6d2
)))4
≤ 12448ρρ+4 log
(
log
(
6d2
))5
log
(
48841/ρρ
)4
≤ 6.5 · 107ρρ+5 log
(
log
(
6d2
))5
This gives the second desired lower bound for h(α), and therefore proves
the lemma. □
Proof of Theorem 1.1. Now, we are ready to prove Theorem 1.1. Suppose,
first, that ρ ≤ log(3d)1/4. By Lemma 3.1, we get
h(α)−1 ≤ g2(ρ, d) ≤ 6.5 · 107
(
log(3d)1/4
)log(3d)1/4+5
log
(
log
(
6d2
))5
≤ 6.5 · 107 exp
((
log(3d)1/4 + 5
)
log
(
log(3d)1/4
)
+ 5 log
(
log
(
log
(
6d2
))))
.
≤ 6.5 · 107 exp
(3
2 log(3d)
1/4 log(log(3d)) + 2 log(3d)1/4 log(log(3d))
)
.
So, we have
(3.1) h(α)−1 ≤ 6.5 · 107 exp
(7
2 log(3d)
1/4 log(log(3d))
)
.
Now, we suppose that ρ ≥ log(3d)1/4. We let r =
⌈
log(3d)1/4
⌉
. In partic-
ular r ≤ ρ. Then, thanks again to Lemma 3.1, we obtain
h(α)−1 ≤ g1(ρ, d) = d1/r
(
1050r5 log(3d)
)r(r+1)2
≤ d1/ log(3d)1/4
(
1050
(
log(3d)1/4 + 1
)5
log(3d)
)(log(3d)1/4+1)(log(3d)1/4+2)2
≤ d1/ log(3d)1/4
(
31693 log(3d)9/4
)6 log(3d)3/4
≤ exp
(
log(3d)3/4 + 6 log(3d)3/4 log
(
31693 log(3d)9/4
))
.
So, we have
(3.2) h(α)−1 ≤ 31693 exp
(
log(3d)3/4 + 272 log(3d)
3/4 log(log(3d))
)
.
476 Jonathan Jenvrin
From (3.1) and (3.2), we obtain
h(α)−1 ≤ 6.5 · 107 exp
(49
2 log(3d)
3/4 log(log(3d))
)
. □
Proof of Corollary 1.2. We fix ϵ > 0. Notice that the function
f(x) = log(log(3x))
log(3x)1/20
has a maximum at
x = e
e20
3 .
Hence, we have that
log(log(3d)) ≤ 20
e
log(3d)1/20,
which implies
(3.3) 492 log(3d)
3/4 log(log(3d)) − ϵ log(d) ≤ 181 log(3d)4/5 − ϵ log(d).
Also, the function fϵ(x) = 181 log(3x)4/5 − ϵ log(x) has its maximum at
x = 13 exp
(724
5ϵ
)5
.
Therefore, we have
(3.4) 181 log(3d)4/5 − ϵ log(d) ≤ 181
(724
5ϵ
)4
− ϵ log(1/3) −
(724
5ϵ
)5
.
Hence, by (3.3) and (3.4), we have that
−492 log(3d)
3/4 log(log(3d)) ≥ −181
(724
5ϵ
)4
+ϵ log(1/3)+
(724
5ϵ
)5
−ϵ log(d).
and we conclude by Theorem 1.1. □
Acknowledgments
I am grateful to Gaël Rémond for comments and suggestions on an earlier
draft of this article, and for pointing out the reference [15]. I am also very
grateful to the anonymous referee for the numerous insightful remarks and
suggestions, which improved the quality of this paper. In particular I thank
the referee for pointing out reference [17] which improved Lemma 2.5.
Lower bounds for the height in Galois extensions 477
References
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Jonathan Jenvrin
100 Rue des Mathématiques, 38610 Gières
France
E-mail: jonathan.jenvrin@univ-grenoble-alpes.fr
URL: https://sites.google.com/view/jonathan-jenvrin-johnny-john/home


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Prime Avoidance Property of $k$ -th Powers of Piatetski–Shapiro Primes