Journal de Théorie des Nombres de Bordeaux

Stéphane Viguié

Invariants and coinvariants of semilocal units modulo elliptic units

Éric Delaygue

Intégralité des coefficients de Taylor de racines d’applications miroir

Daniel Vallières

Connected abelian complex Lie groups and number fields

Kevin J. McGown

Norm-Euclidean Galois fields and the Generalized Riemann Hypothesis

Franz Lemmermeyer

Binomial squares in pure cubic number fields

Paul J. Truman

Hopf-Galois module structure of tame biquadratic extensions

Masami Fujimori

The algebraic groups leading to the Roth inequalities

Melisa J. Lavallee

Blair K. Spearman

Qiduan Yang

Melisa J. LAVALLEE, Blair K. SPEARMAN et Qiduan YANG
PSL(2, 7) septimic fields with a power basis
Tome 24, no 2 (2012), p. 369-375.
<http://jtnb.cedram.org/item?id=JTNB_2012__24_2_369_0>
© Société Arithmétique de Bordeaux, 2012, 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 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 24 (2012), 369-375
PSL(2, 7) septimic fields with a power basis
par Melisa J. LAVALLEE, Blair K. SPEARMAN et Qiduan YANG
Résumé. Nous donnons un ensemble infini de corps de degré 7
monogènes distincts dont la clôture normale a pour groupe de
Galois PSL(2, 7).
Abstract. We give an infinite set of distinct monogenic septimic
fields whose normal closure has Galois group PSL(2, 7).
1. Introduction
Let K denote an algebraic number field of degree n over Q and OK
denote its ring of integers. An integral basis for K is a set {η1, η2, . . . ηn} of
elements of OK such that every element α ∈ OK can be expressed uniquely
in the form
α = x1η1 + x2η2 + · · ·+ xnηn (x1, . . . , xn ∈ Z).
One type of integral basis which is particularly interesting is the power
integral basis. In this case there exists an algebraic integer θ ∈ OK such
that
{
1, θ, θ2, . . . θn−1
}
is an integral basis for K, and the number field K is
said to be monogenic. It is a nontrivial problem to decide whether a number
field has a power basis. The lengthy history of this important problem can
be examined in Gaál [2] and Narkiewicz [9]. For algebraic number fields of
given degree and Galois group of their normal closure, power bases may
be quite rare. As an example, there is only one cyclic quintic field with a
power basis which was shown by Gras [3]. The same phenomenon occurs for
octic fields with Galois group 2 elementary abelian where there is exactly
one field with a power basis as shown by Motoda, Nakahara and Park [6].
On the other hand there are infinitely many dihedral quintic fields with a
power basis which were given by Lavallee, Spearman, Williams and Yang
[5], and infinitely many PSL(2,5) sextic fields with a power basis as was
demonstrated by Spearman, Watanabe and Williams in [8].
In this paper we study septimic field extensions of Q whose normal clo-
sure has Galois group PSL(2, 7), the projective special linear group of 2×2
matrices over F7. We refer to these fields as PSL(2, 7) septimic fields. The
Manuscrit reçu le 11 janvier 2011.
Research supported by the Natural Sciences and Engineering Research Council of Canada.
Mots clefs. Galois Group, Septimic Field, Power Basis.
Classification math. 11R04, 11R32.
370 Melisa J. Lavallee, Blair K. Spearman, Qiduan Yang
purpose of our paper is to show that infinitely many of these septimic fields
are monogenic, as described in the following theorem.
Theorem 1.1. There are infinitely many integers b such that the polyno-
mials
fb(x) = x7 + x6 + x5 + bx4 + (b− 2)x3 − 5x2 − 2x+ 1
define distinct monogenic PSL(2, 7) septimic fields.
2. A parametric family of PSL(2,7) polynomials
For indeterminates a, A we consider the following family of polynomials
due to LaMacchia which can be found in Jensen, Ledet and Yui [4, p. 55].
f(a,A, x) = x7 + 2(1− 3a)x6 + (−3 + 4a+ 8a2)x5 + (−2 + 6a− 14a2)x4
+ (2− 4a+ 6a2 − 8a3)x3 + 8(2 + a)a2x2 + 4(−3 + 2a)a2x
− 8a3 +Ax3(1− x).
The Galois group of f(a,A, x) over Q(a,A) is isomorphic to PSL(2, 7). To
obtain the family of polynomials fb(x) in our theorem we let b be an integer
and choose
a = 1/2, A = b− 5/2,
scale by replacing x by −x and then simplify. Although it is expected that
such a specialization will result in fb(x) being irreducible over Q and having
Galois group PSL(2, 7) this is not guaranteed, so we must confirm the basic
algebraic properties of the polynomials fb(x).
Lemma 2.1. If b is a positive integer then fb(x) is irreducible over Q.
Proof. The polynomials
f0(x) = x7 + x6 + x5 − 2x3 − 5x2 − 2x+ 1
and
f1(x) = x7 + x6 + x5 + x4 − x3 − 5x2 − 2x+ 1
are irreducible modulo 2. Therefore considering the cases b even and b odd
we see that the polynomial fb(x) is irreducible for all integers b. 
Lemma 2.2. If b is an integer then the discriminant of the polynomial
fb(x) is
(2.1) (b2 − 5b− 25)2(27b2 − 135b+ 769)2.
Proof. This calculation was carried out using Maple. 
Lemma 2.3. If b is an integer then not all of the roots of the polynomial
fb(x) are real.
Septimic fields with a power basis 371
Proof. The discriminant of fb(x) is clearly nonzero by (2.1) so that the
roots of fb(x) are distinct. If all of the roots of fb(x) were real then by
Rolle’s Theorem, the nth derivative of fb(x) would have 7−n real roots for
0 ≤ n ≤ 6. However f (5)b (x) = 2520x2 + 720x+ 120 which has no real roots.
This contradiction establishes the result. 
Lemma 2.4. If b is an integer then the Galois group of fb(x) is isomorphic
to PSL(2, 7).
Proof. By Lemma 2.2 the discriminant of the polynomial fb(x) is equal to a
perfect square inQ. A list of possible Galois groups for septimic polynomials
is given by Cohen [1]. Those which correspond to a square discriminant are
C7, the cyclic group of order 7,
M21 or F21 the Frobenius group of order 21,
PSL(2, 7),
A7, the alternating group.
By Lemma 2.3, fb(x) has at least some complex roots. Thus complex con-
jugation is a nontrivial element of the automorphism group of the splitting
field of fb(x) so the order of the Galois group of fb(x) is divisible by 2. This
eliminates C7 and M21. To eliminate A7 we adapt the argument given in
[4, p. 55]. A calculation yields the polynomial identity
(2.2) y3(1 + y)fb(x) + x3(1 + x)f−b+5(y) = p(x, y)q(x, y)
for polynomials p(x, y) and q(x, y) of degree 3 and 4 in x respectively which
are given by
p(x, y) = yx3 + (−y2 + 1)x2 + (y3 + 1)x+ y2 + y
and
q(x, y) = (y3 + y2)x4 + (y4 + 2y3 − y)x3 + (y4 − 3y2 − y + 1)x2
+ (−y3 − y2 − y)x+ y2.
Let β be a root of f−b+5(x). As f−b+5 is irreducible over Q by Lemma 2.1,
β 6= 0,−1. Then setting y = β in (2.2) gives
(2.3) β3(1 + β)fb(x) = p(x, β)q(x, β).
Equation (2.3) is a factorization of fb(x) into factors of degree 3 and 4
respectively over Q(β) which is a degree 7 extension of Q. We deduce from
this that the degree of the splitting field of fb(x) over Q is a divisor of
7 ·3! ·4!. In particular this degree is not divisible by 5 so that the possibility
of A7 as Galois group is now eliminated and the proof is complete. 
372 Melisa J. Lavallee, Blair K. Spearman, Qiduan Yang
3. Calculation of the field discriminant
We have established that the polynomial fb(x) is irreducible over Q and
has Galois group isomorphic to PSL(2, 7). Let θ be a root of fb(x). Then
K = Q (θ) is a PSL(2, 7) septimic field. We set
(3.1) g(b) = g1(b)g2(b)
where
(3.2) g1(b) = b2 − 5b− 25
and
(3.3) g2(b) = 27b2 − 135b+ 769.
We will determine the field discriminant d(K) of K = Q (θ) under the
assumption that g(b) is squarefree. We begin with the following lemma.
Lemma 3.1. Let f(x) = xn+an−1xn−1+· · ·+a1x+a0 ∈ Z[x] be irreducible
over Q. Suppose that α is a root of f(x) and K = Q(α). If p is a prime
number and k a positive integer with k < n such that pk ‖ a0 and pk+1−i | ai,
1 ≤ i ≤ k then the ideal 〈p〉 ramifies in K.
Proof. Suppose that 〈p〉 does not ramify in K. Then there exist distinct
prime ideals ℘1, . . . ℘r in OK such that
〈p〉 = ℘1 · · ·℘r.
As pk ‖ a0 we have as ideals in OK , 〈a0〉 = ℘k1 · · ·℘kr 〈c〉 for some c ∈ Z with
p - c. Thus ℘i - 〈c〉 for i = 1, . . . , r. Since N(α) = ±a0 ≡ 0(mod p) the ideal
〈α〉 must be divisible by at least one ℘i say ℘. As ℘k+1−i | ai 1 ≤ i ≤ k we
have
a0 = a0 − f(α)
= (−a1α− · · · − akαk)− (ak+1αk+1 − · · · − αn) ≡ 0(mod ℘k+1).
since ℘k+1 clearly divides each term inside the pairs of brackets. This con-
tradicts pk ‖ a0. Thus 〈p〉 ramifies in K. 
The next two lemmas determine the ramified primes in Q (θ) . To do this
we give two elements in K and their monic minimal polynomials in Z [x] to
which we can apply the previous lemma. To experimentally find candidates
for these elements we formally solved the polynomial equation fb(x) = 0
for b, calculated b2− 5b− 25 from this, then tried factors of this expression
to find θ1. A similar experiment produced θ2. These elements are
θ1 = −θ4 − 2θ3 − 2θ2 − θ + 1,
and
θ2 = −3θ4 + 2θ2 − 3θ − 3 .
Septimic fields with a power basis 373
Lemma 3.2. Let b ∈ Z be such that g1(b) is squarefree. Let p be a prime
such that p | g1(b). Then p | d(K).
Proof. The element θ1 is clearly a primitive element for K. The monic
minimal polynomial hb(x) of θ1 in Q [x] is
hb(x) = x7 + (1− 6b)x6 + (10b2 + 29b− 40)x5
+ (b4 − 5b3 − 54b2 + 60b+ 475)x4 − (4b2 − 9b− 78)g1(b)x3
+ 3(2b2 − 8b− 45)g1(b)x2 − 4g1(b)2x+ g1(b)2.
As g1(b) is squarefree, Lemma 3.1 applies to the polynomial hb(x) with
k = 2 and we conclude that p | d(K). 
Lemma 3.3. Let b ∈ Z be such that g2(b) is squarefree. Let p be a prime
such that p | g2(b). Then p | d(K).
Proof. The element θ2 is clearly a primitive element for K. The monic
minimal polynomial kb(x) of θ2 in Q [x] is
kb(x) = x7 + 41x6 + (54b2 − 27b+ 318)x5
+ (27b4 − 189b3 + 1066b2 − 782b− 4325)x4
+ (10b2 − 21b− 50)g2(b)x3
+ (36b2 − 114b+ 263)g2(b)x2 + 2g2(b)2x+ g2(b)2.
As g2(b) is squarefree, Lemma 3.1 applies to the polynomial kb(x) with
k = 2 and we conclude that p | d(K). 
Lemma 3.4. If g(b) is squarefree then d(K) = g(b)2.
Proof. Certainly d(K) is a divisor of g(b)2 the discriminant of the polyno-
mial fb(x). Specifically,
g(b)2 = c2d(K)
for some nonzero integer c. On the other hand by Lemmas 3.2 and 3.3
each prime number p dividing g(b) divides d(K). Since g(b) is squarefree
we deduce that p - c. It follows that c2 = 1 proving the Lemma. 
We will require that the polynomial g(b) assume squarefree values for
infinitely many positive integers b. Since the polynomial g(b) is reducible
over Z we can use a proposition due to Nair [7, Theorem C, p. 181-182.].
In order to state this proposition we first define
Nk(f, x, h) = Nk(x, h) = |{n : x < n ≤ x+ h, f(n) is k-free}|
Proposition 3.1. If
f(x) =
m∏
i=1
(fi(x))αi
374 Melisa J. Lavallee, Blair K. Spearman, Qiduan Yang
where each fi is irreducible, α = max
i
αi and deg fi(x) = gi, then
(3.4) Nk(x, h) = Λkh+O
(
h
(log h)k−1
)
for h = xθ where 0 < θ < 1 and k ≥ max{λgi, αi}, (λ =
√
2−1/2) provided
that at least one gi ≥ 2 (the constant Λk is positive).
Lemma 3.5. There exist infinitely many positive integers b such that g(b)
is squarefree.
Proof. The quartic polynomial g(b) is equal to the product of two irre-
ducible quadratic polynomials over Q. The polynomial g(b) has no fixed
square divisors as can be deduced from (3.1), (3.2) and (3.3). To apply the
previous proposition, we note that k ≥ max{λgi, αi} simplifies to k ≥ 2
so that we can set k = 2. The conclusion of the Lemma now follows from
(3.4). 
4. Proof of Theorem
We are ready to prove our theorem.
Proof. The polynomial fb(x) has discriminant equal to
g(b)2 = (b2 − 5b− 25)2(27b2 − 135b+ 769)2,
by Lemma 2.2. Moreover if θ is a root of fb(x) then the field Q (θ) is a
PSL(2, 7) septimic extension field of Q by Lemmas 2.1 and 2.4. By Lemma
3.5 there exist infinitely many positive integers b such that g(b) is squarefree.
For these values of b we have by Lemma 3.4 that the field discriminant of
Q (θ) is equal to g(b)2. It follows that{
1, θ, θ2, θ3, θ4, θ5, θ6
}
is an integral basis for Q (θ) proving that Q (θ) is monogenic. It remains to
show that infinitely many of these monogenic septimic fields are distinct.
This follows from the observation that the discriminant equation
g(b) = ±g(t)
has a finite number of solutions b for a fixed value of t. This completes our
proof. 
References
[1] H. Cohen, A Course in Computational Algebraic Number Theory. Springer-Verlag, 2000.
[2] I. Gaál, Diophantine equations and power integral bases. New Computational Methods.
Birkhauser, Boston, 2002.
[3] M.-N. Gras, Non-monogénéité de l’anneau des entiers des extensions cycliques de Q de
degré premier l ≥ 5. J. Number Theory 23 (1986), 347–353.
[4] C. U. Jensen, A. Ledet, N. Yui, Generic Polynomials, constructive aspects of Galois theory,
MSRI Publications. Cambridge University Press, 2002.
Septimic fields with a power basis 375
[5] M. J. Lavallee, B. K. Spearman, K. S. Williams, and Q. Yang, Dihedral quintic fields
with a power basis. Mathematical Journal of Okayama University, vol. 47 (2005), 75–79.
[6] Y. Motoda, T. Nakahara and K. H Park, On power integral bases of the 2-elementary
abelian extension fields. Trends in Mathematics, Information Center for Mathematical Sci-
ences, Volume 8 (June 2006), Number 1, 55–63.
[7] M. Nair, Power free values of polynomials. Mathematika 23 (1976), 159–183.
[8] B. K. Spearman, A. Watanabe and K. S. Williams, PSL(2,5) sextic fields with a power
basis. Kodai Math. J., Vol. 29 (2006), No. 1, 5–12.
[9] W. Narkiewicz, Elementary and Analytic Theory of Algebraic Numbers. Third Edition,
Springer, 2000.
Melisa J. Lavallee
Department of Mathematics and Statistics
University of British Columbia Okanagan
Kelowna, BC, Canada, V1V 1V7
E-mail: melisa.lavallee@ubc.ca
Blair K. Spearman
Department of Mathematics and Statistics
University of British Columbia Okanagan
Kelowna, BC, Canada, V1V 1V7
E-mail: blair.spearman@ubc.ca
Qiduan Yang
Department of Mathematics and Statistics
University of British Columbia Okanagan
Kelowna, BC, Canada, V1V 1V7
E-mail: qiduan.yang@ubc.ca


PSL$(2,7)$ septimic fields with a power basis

Victor Abrashkin

Ruth Jenni

The field-of-norms functor and the Hilbert symbol for higher local fields

Mark van Hoeij

Vivek Pal

Isomorphisms of algebraic number fields

Not a member yet