Journal de Théorie des Nombres de Bordeaux

David Krumm

Squarefree parts of polynomial values

Mohammad Sadek

Nermine El-Sissi

Mohammad SADEK et Nermine EL-SISSI
Edwards Curves and Gaussian Hypergeometric Series
Tome 28, no 1 (2016), p. 115-124.
<http://jtnb.cedram.org/item?id=JTNB_2016__28_1_115_0>
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Journal de Théorie des Nombres
de Bordeaux 28 (2016), 115–124
Edwards Curves and Gaussian Hypergeometric
Series
par Mohammad SADEK et Nermine EL-SISSI
Résumé. Soit E une courbe elliptique décrite par un modèle
d’Edwards ou un modèle d’Edwards tordu sur Fp, à savoir, E est
définie par une des équations suivantes x2+y2 = a2(1+x2y2), a5−
a 6≡ 0 mod p, ou, ax2 + y2 = 1 + dx2y2, ad(a − d) 6≡ 0 mod p,
respectivement. Nous représentons le nombre de points ration-
nels de E sur Fp en utilisant la série hypergéométrique de Gauss
2F1
(
φ φ
ε
∣∣∣x) où ε et φ sont les caractères trivial et quadratique
sur Fp, respectivement. Cette représentation nous permet d’éva-
luer |E(Fp)| pour certaines courbes elliptiques E, et de démontrer
l’existence d’isogénies entre E et des courbes elliptiques de Le-
gendre sur Fp.
Abstract. Let E be an elliptic curve described by either an
Edwards model or a twisted Edwards model over Fp, namely, E
is defined by one of the following equations x2 + y2 = a2(1 +
x2y2), a5 − a 6≡ 0 mod p, or, ax2 + y2 = 1 + dx2y2, ad(a− d) 6≡ 0
mod p, respectively. We express the number of rational points of E
over Fp using the Gaussian hypergeometric series 2F1
(
φ φ
ε
∣∣∣x)
where ε and φ are the trivial and quadratic characters over Fp
respectively. This enables us to evaluate |E(Fp)| for some elliptic
curves E, and prove the existence of isogenies between E and
Legendre elliptic curves over Fp.
1. Introduction
In [4] Greene initiated the study of Gaussian hypergeometric series over
finite fields. These series are analogous to the classical hypergeometric se-
ries. Several authors managed to find congruence relations satisfied by spe-
cial values of these series, see [6]. Many special values of these series were
determined.
One of the striking aspects of hypergeometric series is that some of their
special values are linked to the number of rational points on some families
Manuscrit reçu le 27 novembre 2013, accepté le 15 mars 2015.
Mathematics Subject Classification. 11T24, 14H52.
Mots-clefs. Edwards Curves, Gaussian Hypergeometric Series, Finite Fields.
116 Mohammad Sadek, Nermine El-Sissi
of algebraic curves over finite fields. Two families of elliptic curves were dis-
cussed in [5]. The number of rational points of an elliptic curve E described
by a Legendre model, namely, y2 = x(x − 1)(x − λ), λ(λ − 1) 6≡ 0 mod p,
over the finite field Fp satisfies the following identity
|E(Fp)| = 1 + p+ pφ(−1) · 2F1
(
φ φ
ε
∣∣∣λ)
where ε and φ are the trivial and quadratic characters over Fp respec-
tively. The latter identity was used to evaluate the hypergeometric series
2F1
(
φ φ
ε
∣∣∣λ) when λ ∈ {−1, 1/2, 2}. If E is defined by a Clausen model,
y2 = (x− 1)(x2 + λ), λ(λ+ 1) 6≡ 0 mod p, then
(1 + p− |E(Fp)|)2 = p+ p2φ(λ+ 1) · 3F2
(
φ φ φ
ε ε
∣∣∣ λ
λ+ 1
)
.
Again this was exploited in order to evaluate the hypergeometric series
3F2
(
φ φ φ
ε ε
∣∣∣ λ
λ+ 1
)
at some specific values of λ.
More Gaussian hypergeometric series appear in formulas describing the
number of rational points on higher genus curves over finite fields. Some of
these formulas can be found in [1] where the following family of algebraic
curves are discussed
yl = x(x− 1)(x− λ), λ(λ− 1) 6≡ 0 mod p, l ≥ 2.
In this note we are interested in elliptic curves described by Edwards
models or twisted Edwards models, namely
x2 + y2 = a2(1 + x2y2), a5 − a 6≡ 0 mod p;
ax2 + y2 = 1 + dx2y2, ad(a− d) 6≡ 0 mod p
respectively.
Edwards models of elliptic curves were proposed in [3] and have been
used since then in many cryptographic applications. The main advantage
enjoyed by these curves is that the group law is simpler to state than on
other models representing elliptic curves. In addition, any elliptic curve
defined over an algebraically closed field k can be expressed in the form
x2 +y2 = a2(1+x2y2). Twisted Edwards models appeared for the first time
in [2] to express more elliptic curves over finite fields with the addition law
being easily formulated.
We show that the number of rational points on an Edwards curve E over
a finite field Fp can be written in terms of the Gaussian hypergeometric
series 2F1
(
φ φ
ε
∣∣∣x). Consequently, we evaluate |E(Fp)| for some E. Then
we prove that every Edwards curve is isogenous to a Legendre curve over
Fp. Finally, it turns out that the number of rational points on a twisted
Edwards Curves and Gaussian Hypergeometric Series 117
Edwards curve E is described using special values of the hypergeometric
series 2F1
(
φ φ
ε
∣∣∣x). This sets the stage for evaluating |E(Fp)| for some
twisted Edwards curves E.
2. Gaussian hypergeometric series
Throughout the note p will be an odd prime unless otherwise stated.
We extend multiplicative characters χ defined over F×p to Fp by setting
χ(0) = 0. We write χ to denote 1/χ. The trivial and quadratic characters
will be denoted by ε and φ respectively. Let J(A,B) denote the Jacobi sum
J(A,B) =
∑
x∈Fp
A(x)B(1− x)
where A and B are characters over Fp. Let A0, A1, . . . , An and B1, . . . , Bn
be characters defined over Fp. The Gaussian hypergeometric series is
n+1Fn
(
A0 A1 . . . An
B1 . . . Bn
∣∣∣x) := p
p− 1
∑
χ
(
A0χ
χ
)(
A1χ
B1χ
)
. . .
(
Anχ
Bnχ
)
χ(x)
where the sum is over all characters over Fp and(
A
B
)
:= B(−1)
p
J(A,B) = B(−1)
p
∑
x∈Fp
A(x)B(1− x).
The following properties of the symbol
(
A
B
)
can be found in [4].
Lemma 2.1. For any characters A and B over Fp, one has:
a) A(1 + x) = δ(x) + p
p− 1
∑
χ
(
A
χ
)
χ(x) where δ(x) = 1 if x = 0 and
δ(x) = 0 if x 6= 0;
b) A(1−x) = δ(x) + p
p− 1
∑
χ
(
Aχ
χ
)
χ(x) where δ(x) = 1 if x = 0 and
δ(x) = 0 if x 6= 0;
c)
(
A
B
)
=
(
A
AB
)
;
d)
(
A
B
)
=
(
BA
B
)
B(−1);
e)
(
A
ε
)
=
(
A
A
)
= −1
p
+ p− 1
p
δ(A) where δ(A) = 1 if A = ε and
δ(A) = 0 otherwise;
118 Mohammad Sadek, Nermine El-Sissi
f)
(
B2χ2
χ
)
=
(
φBχ
χ
)(
Bχ
B2χ
)(
φ
φB
)−1
Bχ(4).
3. Rational points on Edwards curves
Let E be an elliptic curve over a field k with char k 6= 2 defined by an
Edwards model
x2 + y2 = a2(1 + x2y2), where a5 − a 6= 0.
Such an elliptic curve will be called an Edwards curve. If (x1, y1) and (x2, y2)
are two points on E, then these two points add up to
x3 =
1
a
· x1y2 + x2y11 + x1x2y1y2
, y3 =
1
a
· y1y2 − x1x21− x1x2y1y2
.
There are two points at infinity, namely if we homogenize the defining
equation, we get
x2z2 + y2z2 = a2(z4 + x2y2)
and the points at infinity are (x : y : z) ∈ {(1 : 0 : 0), (0 : 1 : 0)}.
We will need the following lemma to count rational points on an Edwards
curve.
Lemma 3.1. Let A be a character on Fp and a ∈ Fp×. The following
identities hold:
a)
(
A2
A
)
=
{(φA
A
)
A(4) if A 6= ε
p−2
p if A = ε
b)
∑
x∈Fp
A(a2 − x2) = pA(4a2)
(
A
2
A
)
=
pA(a2)
(φA
A
)
if A 6= ε
p− 2 if A = ε
Proof. a) In Lemma 2.1 f), put B = ε and χ = A. This yields(
A2
A
)
=
(
φA
A
)(
A
A
)(
φ
φ
)−1
A(4).
According to Lemma 2.1 e), the product above is
(
φA
A
)
A(4) if A 6= ε, and
it is (2− p)
(
φA
A
)
A(4) if A = ε.
To prove b), we notice that∑
x∈Fp
A(a2 − x2) = A
(
4a2
) ∑
x∈Fp
A
(1
2 −
x
2a
)
A
(1
2 +
x
2a
)
Edwards Curves and Gaussian Hypergeometric Series 119
Setting u = 12 −
x
2a , the above sum becomes∑
x∈Fp
A(a2 − x2) = A
(
4a2
) ∑
u∈Fp
A(u)A(1− u)
= A(4a2)J(A,A) = pA(−4a2)
(
A
A
)
.
Using Lemma 2.1 d), one has
(
A
A
)
=
(
A
2
A
)
A(−1). Part b) now follows
from a). 
The following theorem relates the number of rational points on an Ed-
wards curve over Fp to a Gaussian hypergeometric series.
Theorem 3.2. Let E/Fp be described by x2 + y2 = a2(1 + x2y2) where
a5 6≡ a mod p, p is an odd prime. Then
|E(Fp)| = 1 + p+ p · 2F1
(
φ φ
ε
∣∣∣1− a4) .
Proof. The defining equation of E can be written as:
y2 = a
2 − x2
1− a2x2 .
Bearing in mind that there are two points at infinity, one has
|E(Fp)| = 2 + p+
∑
x∈Fp\{±a−1}
φ
(
a2 − x2
1− a2x2
)
= 2 + p+
∑
x∈Fp\{±a−1}
φ
(
1 + a
2 − a−2
a−2 − x2
)
= 2 + p+
∑
x∈Fp\{±a−1}
[
δ
(
a2 − a−2
a−2 − x2
)
+ p
p− 1
∑
χ
(
φ
χ
)
χ
(
a2 − a−2
a−2 − x2
)]
= 2 + p+ p
p− 1
∑
x∈Fp\{±a−1}
∑
χ
(
φ
χ
)
χ
(
a2 − a−2
a−2 − x2
)
= 2 + p+ p
p− 1
∑
χ
(
φ
χ
)
χ
(
a2 − a−2
) ∑
x∈Fp
χ(a−2 − x2).
120 Mohammad Sadek, Nermine El-Sissi
The third equality follows from Lemma 2.1 a). Now we use Lemma 3.1 b)
to obtain the following identity:
|E(Fp)| = 2+p+
p
p− 1
[
p
∑
χ 6=ε
(
φ
χ
)(
φχ
χ
)
χ
(
a−2
)
χ
(
a2 − a−2
)
+
(
φ
ε
)
(p−2)
]
.
Lemma 2.1 d) gives
(
φ
χ
)
=
(
φχ
χ
)
χ(−1) =
(
φχ
χ
)
χ(−1), thus
|E(Fp)| = 2 + p+
p2
p− 1
[∑
χ 6=ε
(
φχ
χ
)(
φχ
χ
)
χ
(
1− a4
)
+ 2− p
p2
]
= 2 + p+ p
2
p− 1
[∑
χ
(
φχ
χ
)(
φχ
χ
)
χ
(
1− a4
)
−
(
φ
ε
)2
+ 2− p
p2
]
= 1 + p+ p · 2F1
(
φ φ
ε
∣∣∣1− a4) .

The Legendre family of elliptic curves is the one defined by
Eλ : y2 = x(x− 1)(x− λ), λ(λ− 1) 6≡ 0 mod p.
Theorem 1 in [5] states that |Eλ(Fp)| = 1+p+p φ(−1)·2F1
(
φ φ
ε
∣∣∣λ). Since
two elliptic curves over Fp are isogenous if and only if they have the same
number of rational points, comparing |Eλ(Fp)| to the number of rational
points of an Edwards curve E over Fp defined by x2 +y2 = a2(1+x2y2), see
Theorem 3.2, yields that E is isogenous to the Legendre elliptic curve E1−a4
if p ≡ 1 mod 4. In fact, every Edwards curve is isogenous to a Legendre
curve over Fp.
Corollary 3.3. Let E be defined by x2 + y2 = a2(1 + x2y2) over Fp, where
p is an odd prime and a5 6≡ a mod p. Then E is isogenous to the Legendre
elliptic curve Ea4 : y2 = x(x− 1)(x− a4).
Proof. According to Theorem 1 in [5],
|Ea4(Fp)| = 1 + p+ pφ(−1) · 2F1
(
φ φ
ε
∣∣∣a4) .
The following identity is Theorem 4.4 (i) of [4]
2F1
(
φ φ
ε
∣∣∣x) = φ(−1) · 2F1 (φ φε ∣∣∣1− x
)
.
Now set x = a4 and use Theorem 3.2. Consequently, E and Ea4 have the
same number of rational points over Fp. It follows that they are isogenous
over Fp. 
Edwards Curves and Gaussian Hypergeometric Series 121
We recall the following Proposition which can be found as Theorem 2
in [5].
Proposition 3.4. Let p be an odd prime. If λ ∈ {−1, 1/2, 2}, then
2F1
(
φ φ
ε
∣∣∣λ) =
0 if p ≡ 3 mod 4,2x·(−1)(x+y+1)/2
p if p ≡ 1 mod 4, x
2 + y2 = p, x is odd.
Corollary 3.5. Let E/Fp be defined by x2 +y2 = a2(1+x2y2) where a5 6≡ a
mod p. If a4 ∈ {−1, 1/2, 2}, then
|E(Fp)| =
{
1 + p if p ≡ 3 mod 4,
1 + p+ 2x ·(−1)(x+y+1)/2 if p ≡ 1 mod 4, x2 + y2 = p, x is odd.
Proof. This follows immediately from Corollary 3.3. 
4. Rational points on twisted Edwards curves
Twisted Edwards curves were introduced in [2] as generalizations of Ed-
wards curves. These curves include more elliptic curves over finite fields
than Edwards curves do. A twisted Edwards curve is an elliptic curve de-
fined by the following twisted Edwards model
ax2 + y2 = 1 + dx2y2
where ad(a − d) 6= 0. For any field k with char(k) 6= 2, any elliptic curve
over k with three k-rational points of order 2 is 2-isogenous over k to a
twisted Edwards curve, see Theorem 5.1 of [2], hence they have the same
number of rational points over Fp.
Lemma 4.1. The following equality holds for any character A over Fp∑
x∈Fp
χ(x2)φ(1− x2) = pφ(−1)
[(
φχ
χ
)
+
(
χ
φχ
)]
.
Proof. We recall that the number of solutions of the equation z2 = a mod
p is given by Na = φ(a) + 1. In fact Na = 2 if a ∈ Fp2, a 6= 0, N0 = 1, and
Na = 0 otherwise. We consider the following sum:∑
x
χ(x)φ(x)φ(1− x) =
∑
x
χ(x)φ(1− x) (Nx − 1)
=
∑
x
χ(x)φ(1− x)Nx −
∑
x
χ(x)φ(1− x)
= 2
∑
x∈F2p
χ(x)φ(1− x)−
∑
x
χ(x)φ(1− x)
=
∑
x
χ(x2)φ(1− x2)−
∑
x
χ(x)φ(1− x).
122 Mohammad Sadek, Nermine El-Sissi
Therefore,
∑
x
χ(x2)φ(1− x2) = J(φχ, φ) + J(χ, φ)
= pφ(−1)
[(
φχ
φ
)
+
(
χ
φ
)]
.
Using Lemma 2.1 c), one has
(
φχ
φ
)
=
(
φχ
χ
)
, similarly
(
χ
φ
)
=
(
χ
φχ
)
. 
Theorem 4.2. Let E/Fp be described by ax2 + y2 = 1 + dx2y2 where
ad(a− d) 6≡ 0 mod p, p is an odd prime. Then
|E(Fp)| = 2 + p+ φ(a) + pφ(−a)
[
2F1
(
φ φ
ε
∣∣∣a−1d)+ 2F1 (φ εφ∣∣∣a−1d
)]
.
In particular,
|E(Fp)| = 2 + p− φ(d) + pφ(−a) · 2F1
(
φ φ
ε
∣∣∣a−1d) .
Proof. We observe that we can write the above twisted Edwards model as
x2 = 1− y
2
a− dy2
.
We set b = a−1d. The number of rational points of E over Fp is given by
|E(Fp)| = 2 + p+
∑
y∈Fp\{
√
ad−1}
φ
(
1− y2
a− dy2
)
= 2 + p+
∑
y∈Fp
φ
(
1− y2
)
φ
(
a− dy2
)
= 2 + p+ φ(a)
∑
y∈Fp
φ(1− y2)φ(1− by2)
= 2 + p+ φ(a)
∑
y∈Fp
φ(1− y2)
[
δ(by2) + p
p− 1
∑
χ
(
φχ
χ
)
χ(by2)
]
.
Edwards Curves and Gaussian Hypergeometric Series 123
The last equality follows from Lemma 2.1 b). According to Lemma 4.1, one
obtains
|E(Fp)| = 2 + p+ φ(a) +
pφ(a)
p− 1
∑
χ
(
φχ
χ
)
χ(b)
∑
y∈Fp
χ(y2)φ(1− y2)
= 2 + p+ φ(a) + p
2φ(−a)
p− 1
∑
χ
(
φχ
χ
)[(
φχ
χ
)
+
(
χ
φχ
)]
χ(b)
= 2 + p+ φ(a) + pφ(−a)
[
2F1
(
φ φ
ε
∣∣∣a−1d)+ 2F1 (φ εφ∣∣∣a−1d
)]
.
However, Corollary 3.16 in [4] indicates that
2F1
(
φ ε
φ
∣∣∣a−1d) = (φ
φ
)
φ(−a−1d)ε(1− a−1d)− 1
p
φ(−1)ε(a−1d)
+ p− 1
p
φ(−1)δ(1− a−1d)δ(ε)
= −1
p
φ(−a−1d)− 1
p
φ(−1).
This proves the theorem. 
Corollary 4.3. Let p be an odd prime. Let E/Fp be defined by ax2 + y2 =
1 + dx2y2 where ad(a− d) 6≡ 0 mod p. If λ := a−1d ∈ {−1, 1/2, 2}, then
|E(Fp)| =

2 + p− φ(d) if p ≡ 3 mod 4,
2 + p− φ(d) + 2x · φ(a) · (−1)(x+y+1)/2
if p ≡ 1 mod 4, x2 + y2 = p, x is odd.
Proof. This follows from Theorem 4.2 and Proposition 3.4. 
We remark that any Legendre curve is isogenous to a twisted Edwards
curve over Fp. Indeed, any elliptic curve E with three Fp-rational points of
order 2 defined by y2 = x(x−a)(x− b) is isogenous to the twisted Edwards
curve 4ax2 + y2 = 1 + 4bx2y2, see Theorem 5.1 in [2]. Thus the formula for
the number of rational points on a Legendre elliptic curve over Fp, Theorem
1 of [5], follows as a special case from Theorem 4.2.
References
[1] R. Barman & G. Kalita, “Hypergeometric functions and a family of algebraic curves”,
Ramanujan J. 28 (2012), no. 2, p. 175-185.
[2] D. J. Bernstein, P. Birkner, M. Joye, T. Lange & C. Peters, “Twisted Edwards curves”,
in Progress in cryptology—AFRICACRYPT 2008, Lecture Notes in Comput. Sci., vol. 5023,
Springer, Berlin, 2008, p. 389-405.
[3] H. M. Edwards, “A normal form for elliptic curves”, Bull. Amer. Math. Soc. (N.S.) 44
(2007), no. 3, p. 393-422 (electronic).
124 Mohammad Sadek, Nermine El-Sissi
[4] J. Greene, “Hypergeometric functions over finite fields”, Trans. Amer. Math. Soc. 301
(1987), no. 1, p. 77-101.
[5] K. Ono, “Values of Gaussian hypergeometric series”, Trans. Amer. Math. Soc. 350 (1998),
no. 3, p. 1205-1223.
[6] R. Osburn & C. Schneider, “Gaussian hypergeometric series and supercongruences”, Math.
Comp. 78 (2009), no. 265, p. 275-292.
Mohammad Sadek
Department of Mathematics and Actuarial Science
American University in Cairo
74 S El-Teseen St, New Cairo
Cairo Governorate 11835
EGYPT
E-mail: mmsadek@aucegypt.edu
Nermine El-Sissi
Department of Mathematics and Actuarial Science
American University in Cairo
74 S El-Teseen St, New Cairo
Cairo Governorate 11835
EGYPT
E-mail: nelsissi@aucegypt.edu


Edwards Curves and Gaussian Hypergeometric Series

Adam Gamzon

Unobstructed Hilbert modular deformation problems

Vincent Pigno

Christopher Pinner

Binomial Character Sums Modulo Prime Powers

Moshe Jarden

Carlos Videla

Moshe JARDEN et Carlos VIDELA
Fields on the Bottom
Tome 28, no 1 (2016), p. 213-219.
<http://jtnb.cedram.org/item?id=JTNB_2016__28_1_213_0>
© Société Arithmétique de Bordeaux, 2016, tous droits réservés.
L’accès aux articles de la revue « Journal de Théorie des Nom-
bres de Bordeaux » (http://jtnb.cedram.org/), implique l’accord
avec les conditions générales d’utilisation (http://jtnb.cedram.
org/legal/). Toute reproduction en tout ou partie de cet article sous
quelque forme que ce soit pour tout usage autre que l’utilisation à
fin strictement personnelle du copiste est constitutive d’une infrac-
tion pénale. Toute copie ou impression de ce fichier doit contenir la
présente mention de copyright.
cedram
Article mis en ligne dans le cadre du
Centre de diffusion des revues académiques de mathématiques
http://www.cedram.org/
Journal de Théorie des Nombres
de Bordeaux 28 (2016), 213–219
Fields on the Bottom
par Moshe JARDEN et Carlos VIDELA
Résumé. Nous notons Qtr le corps des nombres totalement réels.
Etant donné un ensemble S de nombres premiers, nous notons
Q(S) l’extension galoisienne maximale de Q de degré seulement
divisible par des nombres premiers dans S. Nous démontrons que
le corps Qtr,S = Qtr ∩ Q(S) n’a pas de sous corps propre M avec
[Qtr,X : M ] <∞.
Abstract. We denote the field of totally real numbers by Qtr.
For a set S of prime numbers we let Q(S) be the maximal Galois
extension of Q whose degree is divisible only by prime numbers
in S. We prove that the field Qtr,S = Qtr ∩ Q(S) has no proper
subfield M with [Qtr,S : M ] <∞.
Introduction
We say that a field F lies on the bottom if F contains no field E with
1 < [F : E] < ∞. By definition, each of the prime fields Q and Fp lies on
the bottom. By a theorem of Artin, every separably closed field of positive
characteristic lies on the bottom (see for example the proof of [7, Cor. 9.3]).
In particular, the absolute Galois group Gal(K) of a field K of positive
characteristic is torsion free.
The same theorem combined with another theorem of Artin [7, p. 452,
Prop. 2.4] implies that every real closed field lies on the bottom. Again,
this implies that the only torsion elements of the absolute Galois group of
a field K are involutions.
By a theorem of F. K. Schmidt, each Henselian closure of Q with respect
to a prime number p lies on the bottom (e.g. [5, Cor. 15.3]).
By the “Bottom Theorem” [4, Thm. 18.7.7], for every positive integer
e and almost all (σ1, . . . , σe) ∈ Gal(Q)e the field Q̃(σ1, . . . , σe) lies on the
bottom. Here Q̃(σ1, . . . , σe) is the fixed field of σ1, . . . , σe in the algebraic
closure Q̃ of Q. The clause “almost all” means “all but a subset of Gal(Q)e
of Haar measure 0”.
Manuscrit reçu le 27 décembre 2013, révisé le 15 mars 2014, accepté le 11 avril 2014.
Mathematics Subject Classification. 12E30.
Mots-clefs. Bottom theorem, Pythagorean field, totally real number.
Research initiated in a working team during a conference on Definability and decidability
problems in number theory, in the American Institute for Mathematics, Palo Alto, California,
September 2013.
214 Moshe Jarden, Carlos Videla
We mention that Lior Bary-Soroker [1] strengthened the bottom theorem
in the following way: Let K be a finitely generated extension of Q and let
e ≥ 2 be an integer. Then, for almost all (σ1, . . . , σe) ∈ Gal(K)e the field
K̃(σ1, . . . , σe) lies on the bottom [1, Thm. 8.2.2].
Next, we recall that a field F is pythagorean if every sum of two squares in
F is a square in F . It follows that every sum of finitely many squares in F is
a square in F . It also follows that the intersection of pythagorean subfields of
a field Ω (which we assume to be algebraically closed) is pythagorean. Note
that every algebraically closed field is pythagorean. Hence, the intersection
of all pythagorean field extensions of a given field K in Ω is the smallest
algebraic extension of K which is pythagorean. We denote it by Kpyt. If
char(K) 6= 2, then Kpyt is a Galois extension of K. Indeed, Kpyt is the
smallest algebraic extension of K closed under extensions with elements of
the form
√
x2 + y2. By [12, p. 176], Qpyt lies on the bottom.
In order to present our results, we consider the field Qtr of all totally
real algebraic numbers. It is the union of all finite extensions K of Q whose
images under all embeddings into C lie in R. It is also the intersection of
all real closures of Q in Q̃. Since the absolute Galois group of a real closed
field has order two, Gal(Qtr) is generated by involutions.
By a result of Florian Pop ([10] and [11, p. 25]), Qtr is PRC. This means
that every absolutely irreducible variety defined over Qtr with a simple
R–rational point for each real closure R of Qtr has a Qtr–rational point.
This result is a consequence of a local-global principle of Laurent Moret-
Bailly [9, Thm. 1.3 and Rem. 1.7]. Michael Fried, Dan Haran, and Helmut
Völklein proved in [2] that Gal(Qtr) is a free profinite product (in the sense
of Melnikov [8]) of groups of order 2. They also proved that the elementary
theory of Qtr is effectively decidable [3, Thm. 10.1].
Our goal is to enrich the already rich collection of properties of Qtr with
the following one:
Main Theorem. For every set S of prime numbers, the field Qtr,S =
Qtr ∩Q(S) lies on the bottom.
To this end we define K(S) for a field K and a set S of prime numbers as
the union of all finite Galois field extensions L of K whose degrees [L : K]
are divisible only by prime numbers that belong to S.
In particular, if S is the set of all prime numbers, then Q(S) = Q̃ and
Qtr,S = Qtr. In this case, the main theorem becomes the following result.
Corollary. The field Qtr lies on the bottom
The proof of the main theorem uses information about pythagorean
fields, an old theorem of George Whaples, and an older theorem of Ed-
mund Landau.
Fields on the Bottom 215
Remark. As the referee pointed out, if N is a Galois extension of Q,
then the statement “N lies on the bottom” is equivalent to “Gal(N/Q)
is torsion-free”. If N is an arbitrary algebraic extension of Q that lies on
the bottom, then it is still true that Aut(N/Q) is torsion-free. But the con-
verse is not true. For example, Q( 3
√
2) does not lie on the bottom although
Aut(Q( 3
√
2)/Q) is trivial, so torsion-free.
Acknowledgement. The authors are indebted to the referee for comments
that led to an improvement of the presentation of this note.
1. Basic facts
We present a few facts and results that enter the proof of the main
theorem.
Lemma 1.1. If F/M is a cyclic extension of odd prime degree p, then F
has a cyclic extension of degree p.
Proof. By a result of Whaples from 1957 [4, Thm. 16.6.6], M has a Galois
extension N with Gal(N/M) ∼= Zp. The compositum FN is a Galois exten-
sion of F and Gal(FN/F ) ∼= Gal(N/F ∩N). The latter group is isomorphic
to an open subgroup of Zp, hence to Zp itself [4, Lemma 1.4.2]. It follows
that Gal(FN/F ) ∼= Zp. Hence, F has a finite cyclic extension of degree p
in FN . 
Next we need the following result about pythagorean fields which is
proved on page 176 of [12]. It is a corollary of a theorem of Diller and
Dress.
Proposition 1.2. If a finite extension of a field P0 is pythagorean, then
P0 itself is pythagorean.
Finally recall that an algebraic number a is totally real if ϕ(a) ∈ R for
every embedding ϕ: Q̃→ C. If in addition ϕ(a) > 0 for each such ϕ, then a
is totally positive. Note that if a is totally real and a 6= 0, then a2 is totally
positive.
Lemma 1.3. Qtr is a pythagorean field.
Proof. Given elements x, y ∈ Qtr, not both zero, the sum x2 + y2 is totally
positive. Hence, so is z =
√
x2 + y2. Therefore, z ∈ Qtr and x2 + y2 = z2,
as claimed 
Edmund Landau proved in 1919 the following result.
Proposition 1.4 ([6, p. 392, II]). Every totally positive algebraic number
a is a sum of finitely many squares of elements of Q(a).
We mention that two years after Landau published his result, Carl Lud-
wig Siegel improved it by proving that every totally positive algebraic num-
ber a is a sum of four squares in Q(a) [13].
216 Moshe Jarden, Carlos Videla
2. S–Extensions
In addition to the results described in Section 1, we introduce here the
notion of S–extensions and the maximal Galois S–extension of a field.
Let S be a set of prime numbers. An algebraic extensionM/L of fields is
said to be an S–extension, if the degree [M0 : L] of every finite subextension
M0/L of M/L is divisible only by prime numbers that belong to S. In
other words, the degree [M : L] considered as a supernatural number [4,
Remark 22.8.6] is divisible only by prime numbers that belong to S.
For the rest of this section we fix a field K and an algebraic closure K̃
of K. All of the fields that appear in the rest of this section lie between K
and K̃.
Lemma 2.1.
(a) If M/K is an S–extension and L/K is a subextension, then both
M/L and L/K are S–extensions.
(b) If L/K and M/L are S–extensions, then so is M/K.
(c) If M/K is an S–extension and M ′ is a K–conjugate of M , then
M ′/K is an S–extension.
(d) If L/K is an S–extension and L′/K is a Galois S–extension, then
LL′/K is an S–extension.
(e) The compositum of arbitrary family of Galois S–extensions of K is
a Galois S–extension of K.
(f) The compositum of arbitrary S–extensions need not be an S–exten-
sion.
Proof of (a) and (b). Statement (a) follows from the relation [M : L] ·
[L : K] = [M : K] which holds for finite as well as for infinite algebraic
extensions [4, Remark 8.2.6 (4a)]. The same relation proves (b).
Proof of (c). Use that [M : K] = [M ′ : K].
Proof of (d). By (a), L∩L′/K, L/L∩L′, and L′/L∩L′ are S–extensions.
Also, Gal(LL′/L) ∼= Gal(L′/L ∩ L), so [LL′ : L] = [L′ : L ∩ L′], hence
LL′/L is an S–extension. Now apply (b) twice to conclude that LL′/K is
an S–extension.
Proof of (e). We may assume without loss that the family is finite. Then,
we use induction in order to reduce (e) to the case where the family contains
two extensions. The latter case follows from (d).
Proof of (f). Consider S = {3}, let ω be a root of unity of order 3, and let
3√2 be the unique real third root of 2. Then, [Q( 3
√
2) :Q] = 3, [Q(ω 3
√
2) :Q] =
3, but [Q(ω, 3
√
2) : Q] = 6. Thus, Q( 3
√
2) and Q(ω 3
√
2) are S–extensions of
Q but their compositum is not. 
Fields on the Bottom 217
Next we define K(S) to be the union of all finite Galois S–extensions
of K.
Lemma 2.2.
(a) K(S) is a Galois S–extension of K.
(b) Every Galois S–extension of K is contained in K(S).
(c) IfM is an extension of K in K(S), thenM (S) = K(S). In particular,
K(S) has no proper Galois S–extensions.
(d) If 2 ∈ S, then K(S) is pythagorean.
Proof of (a). Consider elements x, x′ ∈ K(S). By definition, there exist fi-
nite Galois S–extensions L and L′ of K that contain x and x′, respectively.
By Lemma 2.1(d), LL′/K is a finite Galois S–extension, so LL′ ⊆ K(S).
It follows that x + x′ and xx′ ∈ K(S) and if x 6= 0 also x−1 ∈ K(S).
Consequently, K(S) is a field. Moreover, K(S) is a Galois S–extension of K.
Proof of (b). IfN/K is a Galois S–extension and x ∈ N , then x is contained
in a finite subextension N0 of N which is Galois over K. By Lemma 2.1(a),
N0/K is an S–extension. Hence, by definition, x ∈ K(S). Therefore, N ⊆
K(S).
Proof of (c). By Lemma 2.1(a), K(S)/M is a Galois S–extension. Hence, it
suffices to prove that if N is a Galois S–extension of M , then N ⊆ K(S).
Indeed, let M ′ be the compositum of all K–conjugates of M . Then,
M ′/K is a Galois extension. By (a), M ′ ⊆ K(S). Moreover, N ′ = M ′N is
a Galois extension of M ′. By Lemma 2.1(d), N ′ is an S–extension of M ,
hence also of M ′ (Lemma 2.1(a)).
Now let N̂ be the compositum of all K–conjugates of N ′. Since M ′/K is
Galois, each of the above conjugates is a Galois extension of M ′. Hence, by
Lemma 2.1(e), N̂/M ′ is an S–extension. Therefore, by Lemma 2.1(b), N̂/K
is a Galois S–extension, so by (b), N̂ ⊆ K(S). It follows that N ⊆ K(S), as
claimed.
Proof of (d). Assume K(S) has elements x, y such that
√
x2 + y2 /∈ K(S).
Then,K(S)(
√
x2 + y2) is a quadratic extension ofK(S), in contrast to (c). It
follows from this contradiction that z =
√
x2 + y2 ∈ K(S) and x2 +y2 = z2,
as desired. 
3. Main Theorem
We fix a set S of prime numbers and recall the main theorem:
Main Theorem. The field Qtr,S = Q(S) ∩Qtr lies on the bottom.
Proof. Assume that Qtr,S does not lie on the bottom. Then, Qtr,S is a finite
proper extension of a fieldM0. Since Qtr,S/Q is Galois, so is Qtr,S/M0. By a
218 Moshe Jarden, Carlos Videla
lemma of Cauchy, Gal(Qtr,S/M0) has an element σ of prime degree p. LetM
be the fixed field of σ in Qtr,S . Then, Qtr,S/M is a cyclic extension of degree
p. By Lemma 2.2(a), Qtr,S/Q is an S–extension, hence by Lemma 2.1(a),
p ∈ S. We distinguish between two cases.
Case A. p 6= 2
By Lemma 1.1, Qtr,S has a cyclic extensionN of degree p. By Lemma 2.2(c),
N ⊆ Q(S). If N 6⊆ Qtr, then NQtr/Qtr is a Galois extension with Galois
group isomorphic to Z/pZ. As mentioned in the introduction, Gal(Qtr) is
generated by involutions. Hence, so is Z/pZ, in contrast to the assumption
p 6= 2. Thus, N ⊆ Qtr. Hence, N = Qtr,S , which is a contradiction.
Case B. p = 2
In this case Qtr,S = M(
√
a) for some non-square element a of M . On the
other hand, by Lemma 2.2(d), Q(S) is a pythagorean field. Since Qtr is
pythagorean (Lemma 1.3), so is the intersection Qtr,S = Q(S) ∩ Qtr. By
Proposition 1.2, M is also pythagorean. Since
√
a ∈ Qtr, the element a
of M is totally positive. By Proposition 1.4, a is a sum of squares in M .
Hence, by the preceding paragraph, a is a square in M , in contrast to the
defining property of a. This ends the proof of Case B and the proof of the
main theorem. 
Corollary 3.1. The field Q(S) lies on the bottom if and only if 2 /∈ S.
Proof. Let R be a real closure of Q and consider a proper Galois extension
N of Q which is not contained in R. Then, N does not lie on the bottom.
Indeed, Gal(N/N ∩ R) ∼= Gal(Q̃/R) ∼= Z/2Z. In particular, if 2 ∈ S, then
Q(
√
−1) ⊂ Q(S), so Q(S) 6⊆ R. Therefore, Q(S) does not lie on the bottom.
On the other hand, if 2 /∈ S, then Q(S) is a quadratic extension of no
subfield (Lemma 2.2(a) and Lemma 2.1(a)). Hence, the argument of the
preceding paragraph proves that Q(S) is contained in every real closure of
Q. Therefore, Q(S) ⊆ Qtr, so Q(S) = Qtr,S . It follows from the main theorem
that Q(S) lies on the bottom. 
References
[1] L. Bary-Soroker, “Pseudo Algebraically Closed Extensions”, PhD Thesis, Tel Aviv Uni-
versity, 2008.
[2] M. D. Fried, D. Haran & H. Völklein, “Absolute Galois group of the totally real num-
bers”, C. R. Acad. Sci. Paris Sér. I Math. 317 (1993), no. 11, p. 995-999.
[3] ———, “Real Hilbertianity and the field of totally real numbers”, in Arithmetic geometry
(Tempe, AZ, 1993), Contemp. Math., vol. 174, Amer. Math. Soc., Providence, RI, 1994,
p. 1-34.
[4] M. D. Fried & M. Jarden, Field arithmetic, third ed., Ergebnisse der Mathematik und ihrer
Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics
and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 11, Springer-
Verlag, Berlin, 2008, Revised by Jarden, xxiv+792 pages.
Fields on the Bottom 219
[5] M. Jarden, “Intersections of local algebraic extensions of a Hilbertian field”, in Generators
and relations in groups and geometries (Lucca, 1990), NATO Adv. Sci. Inst. Ser. C Math.
Phys. Sci., vol. 333, Kluwer Acad. Publ., Dordrecht, 1991, p. 343-405.
[6] E. Landau, “Über die Zerlegung total positiver Zahlen in Quadrate”, Nachrichten von der
Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse 1919
(1919), p. 392-396 (ger).
[7] S. Lang, Algebra, third ed., Graduate Texts in Mathematics, vol. 211, Springer-Verlag, New
York, 2002, xvi+914 pages.
[8] O. V. Mel′nikov, “Subgroups and the homology of free products of profinite groups”, Izv.
Akad. Nauk SSSR Ser. Mat. 53 (1989), no. 1, p. 97-120.
[9] L. Moret-Bailly, “Groupes de Picard et problèmes de Skolem. II”, Ann. Sci. École Norm.
Sup. (4) 22 (1989), no. 2, p. 181-194.
[10] F. Pop, “Fields of totally Σ-adic numbers”, manuscript, Heidelberg, 1992.
[11] ———, “Embedding problems over large fields”, Ann. of Math. (2) 144 (1996), no. 1,
p. 1-34.
[12] P. Ribenboim, L’arithmétique des corps, Hermann, Paris, 1972, 245 pages.
[13] C. Siegel, “Darstellung total positiver Zahlen durch Quadrate”,Math. Z. 11 (1921), no. 3-4,
p. 246-275.
Moshe Jarden
School of Mathematics, Tel Aviv University
Ramat Aviv, Tel Aviv, 6997801
ISRAEL
E-mail: jarden@post.tau.ac.il
URL: http://www.tau.ac.il/~jarden/
Carlos Videla
Mount Royal University
Calgary, Alberta
CANADA
E-mail: cvidela@mtroyal.ca


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