Irrationality of the Angles of Kloosterman Sums over Finite Field

. We prove that the angles of Kloosterman sums over arbitrary ﬁnite ﬁeld are incommensurable with the constant π . In particular, this implies that the Weil bound for Kloosterman sums over ﬁnite ﬁelds cannot be reached.


Introduction
Let F q be the finite field of characteristic p and order q = p m . As usually, denote by F * q the set of non-zero elements of F q , and by ζ n the primitive n−th root of unity e 2πi n . First, for reader's convenience, we recall the notion of classical Kloosterman sum over F q . Definition 1. For each u ∈ F q , the Kloosterman sum K q (u) is a special kind exponential sum defined by and the absolute trace T r(a) over F p of an element a ∈ F q is defined as T r(a) = a + a p + ... + a p m−1 .
It can be easily shown that K q (u) is a real non-zero number. Recall, as well, that the Weil bound (see [14]) states: This inequality implies the existence of a unique real number θ u such that The angle θ u is referred to as angle of the Kloosterman sum K q (u). The behaviour of the angles of Kloosterman sums has been studied by many authors. Here, we only refer to some of these works [1][4] [7][11] [13], and that list of references is definitely far from being complete.
It is worth pointing out the existence of some successful attempts to show that the inequality (1) is always strict for the angles of the simplest Kloosterman sums K p (u), u ∈ F p (see [5,Theorem 8]) which means that θ u = 0, π in this particular case.
In the present article, based on deep facts from Algebraic Number Theory (see, e.g., [9, Chapter 2]), we prove more general result that for any u ∈ F q the ratio θ u /π takes only irrational values, thus establishing additional constraints of the same type as the strictness of the inequality (1). Our result resembles the result with respect to the so-called Frobenius angles obtained in [2] but it seems a transparent logical connection between the two results cannot be found out.
This paper is organized as follows. In the next section, we recall some background from Algebraic Number Theory. Then in Section 3, we give several necessary lemmas. In Section 4, the main result is exposed and illustrated in two examples.

Some Background from Algebraic Number Theory
We need some notions from Algebraic Number Theory (ANT) as algebraic number, minimal polynomial of an algebraic number and algebraic integer (see, e.g., [12,Chapter 3]). An algebraic number is one that satisfies some equation of the form with rational coefficients. (A polynomial having leading coefficient 1 is called monic.) Any algebraic number α satisfies a unique monic polynomial equation of smallest degree, called the minimal polynomial of α, and the algebraic degree of α (over the field of rational numbers Q) is defined as the degree of its minimal polynomial. If an algebraic number α satisfies some equation of type (3) with integer coefficients we say that α is an algebraic integer. The minimal polynomial of an algebraic integer is also with integer coefficients. Also remind that the set of all algebraic numbers forms a number field, i.e. the sum, difference, product and ratio of algebraic numbers are algebraic numbers, too. However, the set of all algebraic integers constitutes only a ring which contains the square root of each own element.
For more sophisticated concepts of ANT we direct the readers to [9,Chapter 2] or [3,Chapter 6]. Herein, in the amount of knowledge needed for this paper, we recall some basic facts concerning those notions.
Let α be an algebraic number with minimal polynomial Evidently, N (α) = (−1) n a n . In general, given a finite extension of number fields L/K, it can be defined the norm N L/K (γ) of an arbitrary γ ∈ L, which in case K = Q and L = Q(γ) coincides with N (γ). (Q(γ) stands for the number field obtained by adjoining γ to Q. In particular, Q(ζ n ) is the so-called cyclotomic field generated by ζ n .) We shall make use of the following properties of norm: , where e is the degree of extension L/Q(α).
Particularly, if α is an algebraic integer then N L/Q (α) ∈ Z.

Several Necessary Lemmas
We also make use of five facts stated hereinafter as lemmata.

Remark 1. Lemma 3 is a part of D. H. Lehmer's work
Also, we need the following obvious lemma.
Lemma 4. The cyclotomic fields Q(ζ k ) and Q(ζ l ) can be embedded in the cyclotomic field Q(ζ m ) where m is the least common multiple of k and l. The next lemma is deduced by a result which is due to M. Moisio and D. Wan (see [10,Lemma 11]), and given below as the following position.
be the minimal polynomial of K q (u). Then, for k = 1, . . . , t, we have .
Lemma 5. For any u ∈ F * q , the absolute norm of Kloosterman sum K q (u) satisfies the congruence: where t is the algebraic degree of K q (u).

Main Result
Now, we are in position to prove the main result of this work.
Theorem 1. All angles θ u of the Kloosterman sums K q (u), u ∈ F q are incommensurable with the constant π.
Proof. By Eq. (2) we have: Assume, on the contrary, θ u = 2πr for some r ∈ Q.
Lemmata 1, 2 and 3 show that the algebraic integers K q (u), √ q = p m/2 and 2 cos 2πr belong to cyclotomic fields. Now, Lemma 4 implies that the number fields Q(K q (u)), Q( √ q) and Q(2 cos 2πr) can be embedded in a common (cyclotomic) field L with extension degrees, say, e 1 , e 2 and e 3 , respectively. Further, on the one hand by P1 and Lemma 5, we easily get: But, on the other hand, by Eq. (4) and properties P2 and P1, we consecutively obtain: N L/Q (K q (u)) = N L/Q ( √ q * 2 cos θ u ) = N L/Q ( √ q)N L/Q (2 cos θ u ) = N e 2 ( √ q)N e 3 (2 cos 2πr).

⊓ ⊔
As an immediate consequence of Theorem 1, we obtain the following corollary.

Corollary 1. The Weil bound cannot be reached by the sums
Proof. Suppose for some u ∈ F q it holds |K q (u)| = 2 √ q. Then, evidently, either θ u = 0 or θ u = π which contradicts the assertion of Theorem 1.

⊓ ⊔
Let us illustrate the claim of Theorem 1 by the following two examples.