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Revista de la Academia Colombiana de Ciencias Exactas, Físicas y Naturales
Print version ISSN 0370-3908
Rev. acad. colomb. cienc. exact. fis. nat. vol.36 no.139 Bogotá Apr./June 2012
MATEMÁTICAS
* Departamento de Matemáticas, Universidad del Valle, Apartado Aéreo 25360, Cali, Colombia E-mail: alvarogr@univalle.edu.co
Universidad Industrial de Santander. Bucaramanga, Colombia. E-mail: ateheran@gmail.com
AMS Classification 2000: 14G05.
Abstract
In this paper we give a generalization of two results obtained by Garcia and Stichtenoth and use them to exhibit a method to construct curves over finite fields whose number of rational points is large compared to their genus. Such curves are induced by algebraic functions fields obtained from elementary abelian -extensions of the rational function field using the trace operator
Key words: Finite Fields, Algebraic Curves, Algebraic Function Fields, Elementary Abelian -Extensions, Rational Points.
Resumen
En este articulo generalizamos dos resultados obtenidos por García & Stictenoth en ([G-S]) y usamos estas generalizaciones para construir curvas sobre cuerpos finitos cuyo número de puntos racionales es grande en comparación con su género. Tales curvas son obtenidas considerando -extensiones abelianas elementales del cuerpo de funciones racionales usando el operador traza .
Palabras clave: Cuerpos finitos, curvas algebraicas, cuerpos algebraicos de funciones, âextensions abelianas elementales, puntos racionales
1. 1. Elementary Abelian -Extensions
Throughout this note we denote by a perfect field of characteristic , by an algebraic function field with constant field and by
the Artin-Schreier operator.
Definition 1.1. For a subset we denote by , the splitting field of all polynomials , with . For such that the extension is called an Artin-schreier extension of .
The following theorem provides a complete description of the Artin-schreier extensions. Its proof depends essentially on the following lemma.
Lemma 1.2. (Hilbert's Theorem 90) Let be a finite extension of whose Galois group G is cyclic generated by . Then has the form for some , if and only if .
Prooƒ: see [L-N] Theorem 2.25.
Theorem 1.3. Let be a field of characteristic . The polynomial
either splits completely over or else, is irreducible over . Moreover the following assertions are equivalent:
Prooƒ: Suppose that , then for we have and since is a separable polynomial of degree , it follows that are all its roots.(1) is a cyclic extension of degree .
(2) , whose minimal polynomial over is , where is defined as (1), for some .
(3) is the splitting field of an irreducible polynomial of the form (1), for some .
Now, its is clear that, if , then is the splitting field of .
It remains to consider the case . Let . To prove that is irreducible over it is enough to prove that , that is to say, that is the minimal polynomial of over (which, from now on, we will denote by min ).
Since is the splitting field of the polynomial we have that is a Galois extension, therefore, it is sufficient to show that .
For this end, observe that since each is completely determined by its action on and permutes all the roots of , then for some , hence, .
Now we will to prove the equivalences:
suppose that is a cyclic extension of degree and let be such that . Since , then by Lemma 1.2, there exist such that . Moreover, since and .
On the other hand, observe that . That is to say,, then therefore, there exist such that and consequently satisfies the polynomial .
Now, since
and then . Thus and (2) ensures that , which is the same as, .
If , with we have that is the splitting field of min .
Assume that is the splitting field of an irreducible polynomial of the form for some . Again by similar arguments as above we obtain that , which means that is cyclic of degree .
Definition 1.4. An extension is said to be an Elementary Ableian -Extension of exponent and degree if is Galois with
The following Theorem states a relationship between the additive subgroups of and the elementary abelian -extensions. To this end, we first need to establish a result.
Theorem 1.5. Let be a field of characteristic >0. There exist a one to one correspondence between the additive subgroups U of containing F which have finite index (U:F), and the elementary abelian -extensions. This correspondence is given by
In such case
The inverse map of is given by
Prooƒ: see {16} page 263
Remark 1.6. observe that, regarding abelian P-torsion group as vector spaces over we can as well define as the map (induced by ) that takes finite-dimensional vector subspaces (over ) of the quotient space to finite dimensional subspaces of the vector space (where F is the inverse image under in some fixed separable closure ). That is to say
Finally if instead of we consider the map
one can see that, one such n-dimensional subspace corresponds, in the notation of Theorem 1.5 to a subgroup with that is a "section" of in the sense that . Therefore, if and are subgroups of such that:
then, the following sentences are equivalent.
(Observe that exactly when ( more accurately, when and are the same subspace of .)) Moreover, if , then
Then, the extension is an elementary abelian -extension of of exponent .
Prooƒ: first observe that since E is the splitting field of the set of polynomials then the extensions is a Galois extensions. On the other hand, since U is an additive subgroup of and char(F) = , then there exists nonzero elements such that
We can find , such that . Now, by Theorem 1.3 we have that
But
So, for , since then
consequently, . Next we prove that For we define as follows,
Observe that is the identity on
Therefore by Theorem 1.3, the polynomial is irreducible over (otherwise and which is a contradiction) hence the are actually well defined. Now it clear that each and , also
In fact, if then,
it follows that i = j and (mod ). Therefore
The converse of the Theorem 1.7 also holds. To prove it, we need the following lemma.
Lemma 1.8. Suppose that L/M is finite Galois extension with Galois Group of the form
where is the coordinate and
then
Prooƒ: see {Ro} Corollary 5.5.4.
Theorem 1.9. If is an elementary abelian -extension of of degree , then
for some additive subgroup U of which satisfies (3).
Prooƒ: since is an elementary abelian extension of degree , then
thus, , where each has order . Let us define for and , be as in the Lemma 1.8 (a)
consequently, by Theorem 1.3. there exist such that for some with , which amounts to, . Observe that and since , from Lemma 1.8 (b), , and therefore
We now claim that are linearly independent over . In fact suppose there is a non-trivial linear combination ,with , then
hence . Now if we assume that then and since, for , we have that , then . On the other hand by Lemma 1.8, , consequently which is a contradiction since . Therefore . Similar arguments will lead us to prove that . Let U be the subgroup generated by . Then by (23), therefore only remains to prove that . In fact, if , then with and each . Therefore it is enough to prove that . For this end, observe that
which is a contradiction.
Theorem 1.10. Let U be an additive subgroup of such that
If , then there exit - intermediate fields such that , where with .
Prooƒ: if , then with and by Theorem 1.3, is either 1 or . but if , then therefore which is a contradiction with the choice of u, hence . On the other hand, if is a subfields such that and then by Theorem 1.3, is the splitting field of one irreducible polynomial of the form - for some . Now, since then by Remark 1.6 we have that hence , for some and , from which ,for some for some . In sum each subfield such that and has the form for some . Finally by Remark 1.6 we obtain the number of these subfields.
Theorem 1.11. Let K be a field of characteristic p > 0 an F/K an algebraic function field of transcendence degree one over K, with constant field K and genus g(F). Consider an elementary abelian extension E/F of degree pn such that K is also the constant field of E. denote by the intermediate fields and by g(E) (resp ) the genus of (resp ). Then
Prooƒ: Let . For a subgroup consider the fixed field whose genus is denoted by and the trace idempotent
Now we shall show that any is contained in precisely t subgroup . In fact, each has the form
where each has order and the set is a basis of over . Now, if , then if and only if for . That is to say, there exist subgroups Hj such that . In other words σ is contained in precisely subgroup of G, therefore
But from and
It follows that, . Thus
Which amounts to, we have the following relation in
The theorem now follows from Kani's result.
Observe that the intermediate extension mentioned in Theorem 1.10 is an Artin-Schreier extension, whose genus, can be computed by [[ST], III.7.8]. This takes us to determine explicitly such intermediate field, which we will call Artin-Schreier intermediate subfields, for which we give the following results generalizing Propositions 1.1 and 1.2 in ([G-S]).
Before that, we should give a definition. We call a polynomial of the specific form
Theorem 1.12. Let be a separable, monic, additive polynomial of degree , with its roots in F. If E/F is an elementary abelian -extension of degree , then there exists an element such that whose minimal polynomial over has the form
Prooƒ: Let us consider the set , it is clear that W is a vectorial space over , moreover W is an additive finite subgroup of F. Now, since each cyclic subgroup of W is order 1 or , then there exist nonzero elements such that
and, the set is a basis of W over . Now from Lemma 1.8 we can choose such that , with . If we define the for as in (16), it is clear that each is an element of order of Gal (E/F) and therefore
On the other hand, since , then σ has a unique representation
With , then the action of σ over the element is given by
where . It follows from (40) that if and only if for k = 1,2,â¦,n, that is, if and only if , and therefore . On the other hand, if and then, since , for some , we have
Reciprocally we have:
Theorem 1.3. Let be a separable, monic, additive polynomial of degree n, with its roots in and such that the polynomial is irreducible over , then the extension where is an elementary abelian -extension of degree n. The intermediate subfield with , have the form where, with and each satisfies the equation therefore , with
Prooƒ: It is clear that is the minimal polynomial for y over . Now, for each then all roots of the polynomial have the form with , and therefore (y) is the splitting field of the polynomial . On the other hand since permutes the roots of then for some , therefore, the application , into group is an isomorphism, that is to say, is an elementary abelian -extension of degree .
Now, if , then and it follow that . On the other hand, since then and therefore
consequently It is say, Now, by Remark 1.6 there exist such subextensions and therefore where . Finally by Theorem 1.10 there exist exactly intermediate fields with therefore such must be one of the .
2. An application to the construction of curves over finite fields
It is well known that algebraic function fields over finite fields have many applications in coding theory, and the latter is closely related to cryptography, see for example [N-Ch]. In this section we exhibit a method to construct algebraic function fields over finite fields (algebraic curves) with many rational places (rational points).
Let be a prime number, the finite field with elements and the rational function field over the finite field . By E/K we mean a function field of transcendence degree one over K , with constant field K. We denote by the maximum number of rational places of the function field E/K of genus g(E/K) = g. The Hasse-Weil bound implies
In 1981 Ihara showed in [1] that
To begin our construction, let us benote by the additive polynomial
Theorem 2.1. The polynomial defined as (47)has the following property:
Proof. It is enough to prove that
Since
for some polynomial , then
Remark 2.2. Observe that in accordance with Theorem 2.1, we have that, for , the equation
The following result provides us a relationship among the genus of the function field E/K and the genus of the Artin-Schreier intermediate subfields .
Theorem 2.3. With the previous notations, the genus of E/K is given by
Remark 2.4. Lemma III.7.7 in [ST], guarantees that we can exclude the case for an integer in the above definition for
Indeed, first observe that from ([ST], i.4.18) together with Theorem 1.11 we have that:
Next, we exhibit a technique that allow us to count the rational places of E/K. for this, we will denote by , the induced curve by the function field E/K.
Lemma 2.5. Let us consider the polynomial
Prooƒ: Let us suppose that exists such that (x, y) belongs to , it is to say . Since , then by taking we obtain the desired result. Reciprocally, if for some and since the function a is surjective ([L-N], 2.23), then for some , it is to say, .
Lemma 2.6. With the above notations, for all .
Prooƒ: By the division algorithm, there exists such that
As consequence of all the above mentioned we have,
Theorem 2.7. The number of rational places of the elementary abelian p-extension defined by (47) is given by
Prooƒ: For fixed , we have that (x, y) is a rational point of the curve if and only if where . In fact, if (x, y) is a rational point of then
Now, for each such that , we have places of degree one of E. Additionally, since for all place of different of , P is unramified in E, then or 0. Therefore,
3. Examples
In this section we give examples of elementary the Abelian p-extensions of the kind given by (47). We will to consider the particular case when n is odd and and we will determine the genus and the number of rational places of these extensions using the formulas (49) and (57).
Example 3.1 It p = 2 and n =3, then k =1, q =8. Also,
Example 3.2. Taking p = n = 3, then q = 27 and k = 1. Also,
The following table contains the values obtained for the genus and the number of rational points by taking different values for p and n, also we compare this values obtained with the Ihara's bound
Acknowledgements. The authors deeply appreciate the helpful comments suggestions made by the referees.
References [ Links ] 1991.
[L-N] Lidl Rudolf and Niederreiter Harald. Introduction to finite fields and their applications. Cambridge university press, [ Links ] 1994.
[Go] V.D. Goppa, Codes on algebraic curves. Sov Math.Dokl 24 (1981), 170-172. [ Links ]
[La] Lang Serge, Algebra, Adisson Wesley Publishing Company, [ Links ] 1970.
[Ka] Kani Ernest, Relations between the genera and between the Hasse-Witt invariants of Galois covering of curves, Canad. Math. Bull, Vol 28, pag 321-327, [ Links ]1985.
[N-Ch] Harald Niederreiter, huaxiong Wang and Chaoping Xing, Function Fields over Finite Fields and their applications to Cryptography, Springer- verlag. [ Links ] 2007
[I] Ihara Y. Some remarks on number of rational points of algebraic curves over finite fields. J Fac Sci Tokyo 28 (1981), p.721-724. [ Links ] [Ro] Roman Steve. Field Theory. Springer-Verlag, [ Links ] 1991
[ST] Stichtenoth Hennig. Algebraic functions fields and codes. Springer- verlag, [ Links ] 1993.
[VV] Van Der Geer Gerard and Van Der Vlugt Marcel. Tables of curves with many points. [Online], http://www.science.uva.nl/ geer. [ Links ]
Recibido el 4 de noviembre de 2009
Aceptado para su publicación el 21 de junio de 2010
Versión revisada recibida el 5 de marzo de 2012