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Axioms 2013, 2(1), 1-9; https://doi.org/10.3390/axioms2010001

Article
On the Content Bound for Real Quadratic Field Extensions
Department of Mathematics/Informatics Institute, Auburn University Montgomery, P. O. Box 244023, Montgomery, AL, USA
Received: 31 October 2012; in revised form: 18 December 2012 / Accepted: 20 December 2012 / Published: 28 December 2012

Abstract

:
Let K be a finite extension of Q and let S = { ν } denote the collection of normalized absolute values on K. Let V K + denote the additive group of adeles over K and let c : V K + R 0 denote the content map defined as c ( { a ν } ) = ν S ν ( a ν ) for { a ν } V K + . A classical result of J. W. S. Cassels states that there is a constant c > 0 depending only on the field K with the following property: if { a ν } V K + with c ( { a ν } ) > c , then there exists a non-zero element b K for which ν ( b ) ν ( a ν ) , ν S . Let c K be the greatest lower bound of the set of all c that satisfy this property. In the case that K is a real quadratic extension there is a known upper bound for c K due to S. Lang. The purpose of this paper is to construct a new upper bound for c K in the case that K has class number one. We compare our new bound with Lang’s bound for various real quadratic extensions and find that our new bound is better than Lang’s in many instances.
Keywords:
adele group; content map; real quadratic extension
Classification:
MSC 11R56; 11R11

1. Introduction

Let K be a finite extension of Q and let S = { ν } denote the collection of normalized absolute values on K. Let V K + denote the additive group of adeles over K and let K + denote the additive group of K viewed as a subgroup of V K + . Let c : V K + R 0 denote the content map defined as c ( { a ν } ) = ν S ν ( a ν ) for { a ν } V K + . We have the following classical result due to J. W. S. Cassels [1](Lemma, p. 66).
Proposition 1.1 
(J. W. S. Cassels) There is a constant c > 0 depending only on the field K with the following property: Let { a ν } V K + be an adele for which c ( { a ν } ) > c . Then there exists a non-zero element b K + V K + for which ν ( b ) ν ( a ν ) , ν S .
Let { c } denote the set of all positive constants for which Proposition 1.1 holds. Then { c } is a non-empty set of real numbers that is bounded below by 0. Thus inf ( { c } ) exists. We define c K = inf ( { c } ) to be the content bound for K. In the case that K is a real quadratic field extension there is a known upper bound for c K due to S. Lang [2](Chapter V, §1, Theorem 0).
Proposition 1.2 
(S. Lang) Let d be a positive square-free integer and let K = Q ( d ) be a quadratic extension.
(i) If d 1 ( mod 4 ) , then c K ( 2 + 2 d ) 2 ,
(ii) if d 2 , 3 ( mod 4 ) , then c K 16 d .
In this paper we construct a new upper bound for c K in the case that K is a real quadratic extension with class number one. We prove the following proposition.
Proposition 1.3 
Let K be a real quadratic extension with class number one. Let f be a fundamental unit of K with f > 1 . Then c K f .
It is of interest to compare our new bound with Lang’s bound for various extensions with class number one. For example, if K = Q ( 86 ) , then the fundamental unit f = 10405 + 1122 86 > 20810 . Since 16 · 86 = 1376 < 20810 , in this case Lang’s bound is better. On the other hand, if K = Q ( 93 ) , then the fundamental unit f = 29 + 3 93 2 < 29 . Since ( 2 + 2 93 ) 2 > 453 , the new bound of Proposition 1.3 is better. Overall, our new bound is better than Lang’s in many instances.
For the convenience of the reader we begin with a review of some preliminary material (§2, §3.) In §4 we prove the formula for our new bound and in §5 we compare our new bound on c K with Lang’s bound for some real quadratic extensions K.

2. Absolute Values

Let K be a finite extension of Q with ring of integers R. An absolute value on K is a function η : K R 0 that satisfies
(i) η ( x ) = 0 if and only if x = 0 ,
(ii) η ( x y ) = η ( x ) η ( y ) , x , y K ,
(iii) there exists a constant M so that η ( 1 + x ) M whenever η ( x ) 1 .
The trivial absolute value is defined as η ( 0 ) = 0 and η ( x ) = 1 for x 0 .
Two absolute values η 1 and η 2 on K are equivalent if there exist r R > 0 so that η 1 ( x ) = ( η 2 ( x ) ) r , x K . Thus, the absolute values on K can be partitioned into equivalence classes. It is well-known that up to equivalence the non-trivial absolute values on Q consist of
| | , | | 2 , | | 3 , | | 5 , | | 7 , .
Here | | is the ordinary absolute value on R restricted to Q , and for a rational prime p, | | p is the p-adic absolute value defined as | 0 | p = 0 and
| x | p = 1 p m
for x = ( r / s ) p m , ( r , p ) = ( s , p ) = 1 , m Z .
Let η be an absolute value on K. Then η determines a topology on K where the basic open sets are of the form U x , ϵ , x K , ϵ > 0 , with
U x , ϵ = { y K : η ( x - y ) < ϵ }
The topology thus described is the η-topology on K. Let K η denote the completion of K with respect to the η-topology. In a natural way the absolute value η on K extends to a unique absolute value on K η , which we also denote by η, cf. [3](Chapter XII, §2). In the case K = Q , η = | | , the completion Q | | is the set of real numbers R . If K = Q , η = | | p , then the completion Q | | p is the field of p-adic rationals, Q p . If L is a finite extension of the completion K η , then the absolute value η on K η extends uniquely to an absolute value η * on L and L is complete with respect to the η * -topology [3](Chapter XII, Proposition 2.5).
If K is a finite extension of Q of degree N, then each absolute value on Q extends to a finite number ( N ) of absolute values η on K[1](Chapter II, Theorem, p. 57). To see how the ordinary absolute value | | extends to K, let K = Q ( α ) for some α C , and let p ( x ) = irr ( α ; Q ) . Let p ( x ) = i = 1 g p i ( x ) denote the factorization of p ( x ) over R into irreducible polynomials. Note that g N . For each i, 1 i g , there exists an embedding λ i : K R ( α i ) , α α i , where α i is a root of p i ( x ) . One defines an absolute value η i on K by setting
η i ( x ) = | λ i ( x ) | * , x K
where | | * is the unique extension of | | to R ( α j ) . The collection η 1 , η 2 , , η g is the set of extensions of | | to K.
The p-adic absolute value | | p extends to K in the following manner. Let ( p ) = P 1 e 1 P 2 e 2 P g e g be the unique factorization of ( p ) into prime ideals P i of R. Each P i corresponds to an extension η i of | | p to K as follows. Put η i ( 0 ) = 0 . For r R , r 0 , let t r be the integer t r 0 for which ( r ) P i t r , ( r ) P i t r + 1 . Now let x = r / s K , r 0 , s 0 . One then puts
η i ( x ) = 1 p ( t r - t s ) / e i
The collection η 1 , η 2 , , η g is the set of extensions of | | p to K. Since g N , there are at most N extensions.
The extensions η of | | are the Archimedean absolute values on K. The extensions of η of | | p are the non-Archimedean (or discrete) absolute values on K. Absolute values η on K obtained as extensions constitute all of the absolute values on K (up to equivalence.)
If η i is Archimedean and corresponds to a real embedding λ i , then the local degree d η i = [ R ( α i ) : R ] = 1 , and we define the normalized absolute value to be
ν i ( x ) = η i ( x ) , x K
If η i is Archimedean and corresponds to a complex embedding λ i , then the local degree d η i = 2 , and we define the normalized absolute value as
ν i ( x ) = ( η i ( x ) ) 2 , x K
If η i is a discrete extension of | | p corresponding to the prime ideal P i , the local degree is d η i = e i f i where f i = [ R P i / P i R P i : F p ] is the residue class field degree. In this case the normalized absolute value is given as
ν i ( x ) = ( η i ( x ) ) e i f i = 1 p ( t r - t s ) / e i e i f i = 1 p ( t r - t s ) f i
where x = r / s .
If ν is the normalized absolute value obtained from η, then the ν-topology on K is equal to the η-topology on K since ν and η are equivalent absolute values. In what follows we let S = { ν } denote the set of normalized absolute values on K; K ν denotes the completion of K with respect to the ν-topology. For ν discrete, we let R ν denote the ring of integers in K ν . The absolute value ν extends to an absolute value on K ν (also denoted by ν.) We consider K ν to be endowed with the ν-topology.

3. The Adele Ring

Let K be a finite extension of Q and let S = { ν } denote the set of normalized absolute values on K. For each discrete ν, R ν is a compact open subset of K ν . The adele ring V K over K is the topological ring that is the restricted product of the completions K ν with respect to the collection { R ν : ν discrete } , together with the restricted product topology on the completions K ν with respect to the collection { R ν : ν discrete } . This means that V K consists of those vectors
{ , a ν , } ν S K ν
for which a ν R ν for all but finitely many ν. The ring structure of V K is given component-wise:
{ , a ν , } + { , b ν , } = { , a ν + b ν , }
{ , a ν , } · { , b ν , } = { , a ν b ν , }
We write { a ν } for the adele { , a ν , } . A basis for the topology on V K consists of open sets of the form
ν S U ν
where U ν is open in K ν for all ν and U ν = R ν for all but finitely many ν.
Let V K + denote the additive group of the adele ring V K and let K + denote the additive group of K.
Proposition 3.1 
Let b K + , b 0 . Then ν S ν ( b ) = 1 .
Proof. For two proofs, see [1](Chapter II, Theorem, p. 60 and p. 66).                  ⋄
Proposition 3.2 
K + embeds into V K + through the map b { b , b , b , } .
Proof. Let b K + and write b = a / c , where a , c R , c 0 . Since there are only a finite number of prime divisors of c, c - 1 R ν for all but a finite number of ν. Thus { b ν } with b ν = b for all ν S is an adele of K. It is easy to show that the map b { b , b , b , } is an injection of groups K + V K + .      ⋄
With these preliminaries in mind, we now give two upper bounds for the content bound c K in the case that K is a real quadratic extension with class number one.

4. Two Bounds for c K

Let d be a square-free positive integer, let K = Q ( d ) denote the real quadratic extension with ring of integers R. Let c K be the content bound for K. We recall some number-theoretic facts about K. If d 1 ( mod 4 ) then R = Z [ 1 + d 2 ] and if d 2 , 3 ( mod 4 ) then R = Z [ d ] . The discriminant disc ( R ) = d if d 1 ( mod 4 ) , and disc ( R ) = 4 d if d 2 , 3 ( mod 4 ) . If d 1 ( mod 4 ) , then the only rational primes that ramify are divisors of d. If d 2 , 3 ( mod 4 ) , the rational primes that ramify are 2 and the divisors of d.
The set of normalized absolute values on K is computed as follows. The Archimedean absolute value | | on Q extends to two normalized absolute values, ρ 1 , ρ 2 , defined as follows. For a + b d K ,
ρ 1 a + b d = a + b d
and
ρ 2 a + b d = a + b d ¯ = a - b d
The discrete absolute values on Q extend to K in the following manner. If p disc ( R ) , then ( p ) = P 2 for some prime ideal P of R. Thus | | p extends to one normalized absolute value ν on K. On the other hand, if p disc ( R ) and ( d p ) = - 1 , then ( p ) = P for P prime, and so, p remains prime in R. In this case, | | p extends to one normalized absolute value ν on K. If p disc ( R ) and ( d p ) = 1 , then ( p ) = P Q for P , Q prime and so, | | p extends to two normalized absolute values ν , ν .
Let S = { ν } denote the set of normalized absolute values on K, and let V K + be the additive group of adeles. There is a known bound for c K due to S. Lang [2].
Proposition 4.1 
(S. Lang) Let d be a positive square-free integer and let K = Q ( d ) be a quadratic extension.
(i) If d 1 ( mod 4 ) , then c K ( 2 + 2 d ) 2 ,
(ii) if d 2 , 3 ( mod 4 ) , then c K 16 d .
Proof. For a proof see [2](Chapter V, §1, Theorem 0).                        ⋄
To prove our formula for a new bound on c K , we need some lemmas regarding units in R. The units group of R is
- 1 × h
where h is a fundamental unit in R. Note that h R .
Lemma 4.2 
There exists a fundamental unit f in R with f > 1 .
Proof. Let h be a fundamental unit. We consider first the case h > 0 . If h > 1 , then we set f = h and condition is satisfied. Else, assume that 0 < h < 1 . Then h h - 1 = 1 implies that h - 1 > 1 . Of course, h - 1 is a fundamental unit and so we set f = h - 1 . If h < 0 , then - h > 0 is a fundamental unit and as shown above we may take f > 1 .                          ⋄
Lemma 4.3 
If h is a fundamental unit of R, then h ¯ = ± h - 1 .
Proof. Let N K / Q : K Q be the norm map defined as
N K / Q a + b d = a + b d a + b d ¯ = a 2 - b 2 d
for a , b Q . The norm map restricts to give a map N K / Q : R Z . Now suppose that h is a fundamental unit with inverse h - 1 . Then N K / Q ( h ) and N K / Q ( h - 1 ) are in Z . Moreover, h h - 1 = 1 yields
1 = N K / Q ( h h - 1 ) = N K / Q ( h ) N K / Q ( h - 1 )
Consequently, N K / Q ( h ) = ± 1 , and thus h h ¯ = ± 1 , or h ¯ = ± h - 1 .                    ⋄
We now give the new bound on c K .
Proposition 4.4 
Let d be a square-free positive integer, let K = Q ( d ) and assume that K has class number one. Let f > 1 be a fundamental unit in R. Then c K f .
Proof. We show that if { a ν } is an adele in V K + with c ( { a ν } ) f , then there exists b K + , b 0 , so that ν ( b ) ν ( a ν ) for all ν S . For ν discrete, let K ν denote the completion of K with respect to the ν-topology, and let R ν denote the ring of integers in K ν . We have
a ν = u ν π ν m ν
where u ν is a unit in R ν , m ν Z , and where π ν is a uniformizing parameter for R ν . Since { a ν } is an adele, m ν 0 for all but a finite number of ν, and since ν S ν ( a ν ) > 0 , m ν = 0 for all but a finite number of ν. Let ν 1 , ν 2 , , ν k denote the collection of discrete ν for which m ν i 0 , listed so that ν 1 , ν 2 , , ν l are those ν i with m ν i > 0 and ν l + 1 , ν l + 2 , , ν k are the ν i for which m ν i < 0 .
For i = 1 , 2 , , k , let P ν i denote the ideal of R corresponding to the discrete normalized absolute value ν i . Then the ideal
P ν 1 m ν 1 P ν 2 m ν 2 P ν l m ν l
is principal and generated by an element α R . Moreover, the ideal
P ν l + 1 - m ν l + 1 P ν l + 2 - m ν l + 2 P ν k - m ν k
is principal and generated by an element β R .
Let Λ = α / β . We can assume without loss of generality that Λ > 0 . For all ν discrete, ν ( a ν ) = ν ( Λ ) . Thus
f ν S ν ( a ν ) = ρ 1 ( a ρ 1 ) ρ 2 ( a ρ 2 ) ν dis. ν ( Λ )
Since f > 1 , there exists an integer k for which
f k ρ 1 ( a ρ 1 ) Λ < f k + 1
Put b = f k Λ . Then ρ 1 ( b ) ρ 1 ( a ρ 1 ) . Now from Inequality (2),
f ¯ k ρ 1 ( a ρ 1 ) Λ < f ¯ k f k + 1 = ( | ± f - 1 | ) k f k f , by Lemma 4.3 = ( | ± f - 1 f | ) k f = f
So,
f ¯ k ρ 2 ( Λ ) < f Λ ρ 1 ( a ρ 1 ) ρ 2 ( Λ ) ρ 1 ( a ρ 1 ) ρ 2 ( a ρ 2 ) ν dis. ν ( Λ ) Λ ρ 1 ( a ρ 1 ) ρ 2 ( Λ ) , by (1) = ρ 2 ( a ρ 2 ) ρ 1 ( Λ ) ρ 2 ( Λ ) ν dis. ν ( Λ ) , since ρ 1 ( Λ ) = Λ = ρ 2 ( a ρ 2 ) ν S ν ( Λ ) = ρ 2 ( a ρ 2 ) , by Proposition 3.1 .
Observe that
f ¯ k ρ 2 ( Λ ) = ( ρ 2 ( f ) ) k ρ 2 ( Λ ) = ρ 2 ( f k ) ρ 2 ( Λ ) = ρ 2 ( f k Λ ) = ρ 2 ( b )
thus ρ 2 ( b ) < ρ 2 ( a ρ 2 ) . For ν discrete, f k is a unit in R ν . Thus ν ( b ) = ν ( Λ ) = ν ( a ν ) for all ν discrete. Thus b is as required.                          ⋄

5. A Comparison of Bounds

Let d be a square-free positive integer, let K = Q ( d ) and let c K be the content bound for K. In Table 1 below we compare Lang’s bound on c K (Proposition 4.1) with the new bound obtained in Proposition 4.4 for all d < 50 for which K has class number one. The values of d were obtained from [4](A003172). The values of the fundamental unit f were computed using an algorithm based on [5](Theorem 11.5) as implemented in [6]. We conclude that our new bound is better than Lang’s bound in 17 out of 22 of the cases.
Of course, the fundamental unit f has been proven to be a bound for c K only in the case that K has class number one. It would be of interest to extend Proposition 4.4 to the case where K has class number greater than one.
Table 1. Lang’s bound compared with the fundamental unit.
Table 1. Lang’s bound compared with the fundamental unit.
dLang’s bound (L)new bound (f)L vs. f
232 1 + 2 f 2 . 41421
348 2 + 3 f 3 . 73205
5 ( 2 + 2 5 ) 2 41 . 8885 1 + 5 2 f 1 . 61803
696 5 + 2 6 f 9 . 89898
7112 8 + 3 7 f 15 . 9373
11176 10 + 3 11 f 19 . 9499
13 ( 2 + 2 13 ) 2 84 . 8444 3 + 13 2 f 3 . 30278
14224 15 + 4 14 f 29 . 9666
17 ( 2 + 2 17 ) 2 104 . 985 4 + 17 f 8 . 12311
19304 170 + 39 19 339 . 997 L = 304
21 ( 2 + 2 21 ) 2 124 . 661 5 + 21 2 f 4 . 79129
22352 197 + 42 22 393 . 997 L = 352
23368 24 + 5 23 f 47 . 9792
29 ( 2 + 2 29 ) 2 163 . 081 5 + 29 2 f 5 . 19258
31496 1520 + 273 31 3040 L = 496
33 ( 2 + 2 33 ) 2 181 . 957 23 + 4 33 f 45 . 9783
37 ( 2 + 2 37 ) 2 200 . 662 6 + 37 f 12 . 0828
38608 37 + 6 38 f 73 . 9865
41 ( 2 + 2 41 ) 2 219 . 225 32 + 5 41 f 64 . 0156
43688 3482 + 531 43 6964 L = 688
46732 24335 + 3588 46 48670 L = 732
47752 48 + 7 47 f 95 . 9896

References

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