arXiv:2005.00911v1 [math.NT] 2 May 2020 felements of e hoe 1o [ of 11 Theorem see m ( β oegnrlfruahls if inclusion-exclusion holds: simple formula a general through more obtained is elements h nso maximal of ones the h oi oyoilin monic the oyoil eoe by denoted polynomial, ftecnuae of conjugates the of sa da of ideal an is element hscnet nelement an context, this for lmnsi nw o arbitrary for known is elements n,frec each for and, o hi ubr fΦ If F . their for group eosrethat observe We sflos h field The follows. as h atrcnb bandvathe via obtained be can latter the elements x q α,q ∈ n OEO DIIECAATR FFNT FIELDS FINITE OF CHARACTERS ADDITIVE ON NOTE A [ Let e od n phrases. and words Key 2010 Date exist the ensures (PNBT) Theorem Basis Normal Primitive The x g − F ,teeeitΦ exist there ], ( divides q x 1) n a ,2020. 5, May : F F = ) ahmtc ujc Classification. Subject Mathematics h rmtv omlBssTerm.I hsnt eprovide we note objects. over this two results In these existence between Theorem). of relation Basis proof Normal the Primitive in the used extensively been The n,frec integer each for and, Abstract. q ∗ snra over normal is q ◦ α n β α hsrsl a rtpoe yLntaadSho [ Schoof and Lenstra by proved first was result This . etefiiefil with field finite the be ∈ ∈ P F α = x q F odr fa lmn in element an of -orders F ∈ i m n q F α =0 q n − q n q F h set the , [ n a .The 1. x q F htaepiiie .. eeaoso h ylc(multipl cyclic the of generators i.e., primitive, are that Let n − i ,hnepicpl.The principal.. hence ], q x 7 q n with i α n e . f[ of 2.2 Sec and ] F ( β F F x ehv that have we , ∈ a evee san as viewed be can q n q F n q odr h oml Φ formula The -order. diiecaatr,fiiefields, finite characters, additive opie an comprises q n etefiiefil with field finite the be q m F F − ≥ I β m F eoe h ue oin ucinfrplnmasin for function totient Euler the denotes fadol if only and if n q sedwdwt h following the with endowed is q α α,q odrraiypoie rtro o omlelements: normal for criterion a provides readily -order [ q 1) α,q x ∈ ≥ [ ,let 1, ftepolynomials the of x and ] htgnrtsteideal the generates that , flatdge uhthat such degree least of ] F ,let 1, > = f 1. q n ∈ q f omleeet of elements normal 0 Introduction snra over normal is UA REIS LUCAS F F and F q α qasΦ equals F F q q q q q n x n n lmns where elements, [ ∈ 9 F odro lmnsin elements of -order 22 piay,11T06(secondary). (primary), 12E20 x n n nadtv hrce over character additive an and ]. q eteunique the be samncdvsrof divisor monic a is ] m eteunique the be n − 1 F bssfor -basis β,q q n,i at ehv lsdformula closed a have we fact, in and, 1 n q eset we , q I ∈ ( F F lmns where elements, ( x q f q q = ) odrof -order ( vco pc fdimension of space -vector .Frmr eal nteefacts, these on details more For ). α x g n ( o every for F x x F − F n q ) q n q g n )frtenme fnormal of number the for 1) − odr eirclo polynomials. of reciprocal -order, ∈ dge xeso of extension -degree fteset the if h xsec fnormal of existence The . ◦ q n .S omleeet are elements normal So 1. F I F α dge xeso of extension -degree α q α = F q m [ Equivalently, . n x q = q α ste(nqe monic (unique) the is nt ed (e.g., fields finite α,q o which for ] ruet nfc,a fact, In argument. [ F sapiepower prime a is p x over ∈ q P s -ouestructure: ]-module x ninteresting an n ( x n F hc sdefined is which , sapiepower prime a is { i m =0 ) ,β β, q − n F F ◦ 5 neo normal of ence nparticular, in ; q ,tenumber the 1, q a n proof a and ] α n ro of proof A . i q α β , . . . , have .Since 0. = q F g i q o an For . ◦ . icative) m α n q α,q In . n 0, = − F 1 q is } . 2 LUCAS REIS without any use of computers was later given by Cohen and Huczynska [2]. The main idea is to use characters of Fqn to build characteristic functions for the set of normal and primitive elements. This idea has been extensively used in the proof of the existence of elements in finite fields with many specified properties, beyond normality and primitivity. For more details see, Chapter 3 of [1] and the references therein. The characteristic function for the set of normal elements is obtained via ad- s ditive characters. Write q = p , where p is the characteristic of Fq. An additive × character of Fqn is a function χ : Fqn → C such that χ(a + b)= χ(a) · χ(b). By × the definition, each a ∈ Fqn induces the additive character χa : Fqn → C with

2πi · Trqn/p(aα) χ (α) = exp , a  p  ns−1 pi F F where Trqn/p(x) = i=0 x denotes the trace of qn on p. It is well known that every additiveP character of Fqn is of such form. Moreover, from the identity of mappings χa · χb = χa+b, we have that the set Fqn of additive characters of F n is an abelian group (written multiplicatively), isomorphic to F n ; the identity q d q element is the trivial character χ0 with χ0(α) = 1 for every α ∈ Fqn . The previous Fq[x]-module structure of Fqn lifts to the following Fq[x]-module structure on Fqn : for g ∈ Fq[x] and χ ∈ Fqn , we set g ◦ χ : α → χ(g ◦ (α)), which F is another elementd of qn . Within this structure,d we have a natural extension of F -order to additive characters: the set I of the polynomials g(x) ∈ F [x] for q d χ q which g ◦ χ = χ0, is an ideal of Fq[x], hence principal. The Fq-order of χ is the (unique) monic polynomial, denoted by Ord(χ), that generates the ideal Iχ. It is n direct to verify that, as before, Ord(χ) ∈ Fq[x] is a divisor of x − 1. In the proof of the PNBT [2, 5], the character-sum formula for the characteristic function of normal elements depends on the sets

Cf,q := {χ ∈ Fqn | Ord(χ)= f}, n F where f runs over the monic divisorsd of x − 1 in q[x]. We observe that the elements of Cf,q are of the form χa with a ∈ Fqn . Although no explicit description of the sets Cf,q is required in the proof of the PNBT, further work needed to describe such sets for special values of f. For instance, see [3], where the cases f(x)=1 and f(x)= x − 1 are considered, and the sets C1,q = {χ0} and Cx−1,q = F∗ {χa | a ∈ q} are obtained. The aim of this note is to provide the connection between the Fq-orders of an element α ∈ Fqn and its associated additive character χα ∈ Fqn . Our result is stated as follows. F F m i Theoremd 1.1. Let α ∈ qn be an element of q-order f(x) = i=0 aix . Then ∗ −1 m the Fq-order of χα equals f (x)= a0 x f(1/x), the monic reciprocalP of f. As an immediate consequence of the previous theorem we have that, for each n monic divisor f ∈ Fq[x] of x − 1, the following holds: ∗ Cf,q = {χa | ma,q = f }. F∗ In particular C1,q = {χ0}, Cx−1,q = {χa | a ∈ q} and Cxn−1,q comprises the characters χβ with β ∈ Fqn a normal element over Fq. ANOTEON ADDITIVECHARACTERS OFFINITE FIELDS 3

2. Proof of Theorem 1.1

We observe that, for monic polynomials f, g ∈ Fq[x] with gcd(f(x),x) = gcd(g(x),x) = 1, we have the identity (f ∗)∗ = f and f divides g if and only ∗ ∗ if f divides g . Moreover, the Fq-order of elements in Fqn and Fqn are divisors of xn − 1, hence relatively prime with x. From these facts it suffices to prove d that, for every α ∈ Fqn and every monic g ∈ Fq[x] of degree at most n − 1 with ∗ gcd(g(x),x) = 1, g ◦ α = 0 if and only if g ◦ χα = χ0. s m m−1 i F Write q = p and g(x)= amx + i=0 aix ∈ q[x], where m

2πi · Trqn/p(α · Lg(a)) g ◦ χ (a) = exp , α  p  qm m−1 qi ∗ where Lg(x)= x + i=0 aix . Therefore, it suffices to prove that g ◦ α = 0 if and only if P qn Trqn/p(αLg(x)) ≡ 0 (mod x − x). n Let M(x) ∈ Fq[x] be the unique polynomial of degree at most q − 1 such that qn Trqn/p(αLg(x)) ≡ M(x) (mod x − x). It is clear that M(x) is of the form ns−1 pi F F p i=0 cix . Since Trqn/p(a) ∈ p for every a ∈ qn , we have that M(x) ≡ M(x) qn p P(mod x − x). In particular, cj = cj−1 for every 1 ≤ j ≤ n. Therefore, M(x) vanishes if and only if c0 = 0. Let us compute the coefficient c0. We observe that pi ns−1 m−1 m−1 ns−1 qj pi pi psj+i Trqn/p(αLg(x)) = α ajx  = aj α x . Xi=0 Xj=0 Xj=0 Xi=0   psn qn Since x ≡ x (mod x − x), in the last sum, the terms contributing to c0 are pi pi pi+sj the ones of the form aj α x with i + sj ≡ 0 (mod sn), where 0 ≤ i ≤ sn − 1 and 0 ≤ j ≤ m

3. Additional Remarks

A t-degree monic polynomial f ∈ Fq[x] is self-reciprocal if it coincides with its monic reciprocal, i.e., ∗ − f (x)= f(0) 1xtf(1/x)= f(x). The following corollary is an immediate consequence of Theorem 1.1. 4 LUCAS REIS

Corollary 3.1. The Fq-orders of α ∈ Fqn and its associated additive character χα ∈ Fqn coincide if and only if one of them is self-reciprocal. Ind Subsec. 2.3 of [4] and Subsec. 2.3.1 of [8], the authors mistakenly claimed that the Fq-orders of α and its associated character χα coincide for every α ∈ Fqn . Fortunately, the results on these papers are not affected by this false claim. As in the proof of the PNBT, the results of [8] do not require explicit description of sets Cf,q (see Subsec. 3.1 of [8]). In particular, the claim relating the Fq-orders of α and χα are not employed in [8]. Moreover, in [4], this claim is implicitly used only m for characters of Fq-order 1 and (x − 1) , where m is a power of the characteristic p (see Theorem 3.1 in [4]). The latter comprise only self-reciprocal polynomials, where the claim is true by Corollary 3.1. Applying the previous corollary, we characterize the positive n for which the false claim in [4, 8] holds true. We have seen that the Fq-orders of α n and χα are divisors of x − 1. Conversely, for each monic divisor f ∈ Fq[x] of n x − 1, there exist elements in Fqn with Fq-order f. Hence the Fq-orders of α and n χα coincide for every α ∈ Fqn if and only if every monic divisor of x − 1 over Fq is a self-. Equivalently, every monic irreducible divisor of v u x − 1 over Fq is self-reciprocal, where n = p · v with gcd(v,p) = 1. According to Theorem 1 of [6], the latter holds if and only if qj ≡−1 (mod v) for some positive integer j ≤ v. We obtain the following corollary. u Corollary 3.2. Write n = p ·v, where gcd(v,p) = 1. The Fq-orders of α and χα j coincide for every α ∈ Fqn if and only if q ≡−1 (mod v) for some 1 ≤ j ≤ v. References [1] P. Charpin, A. Pott, A. Winterhof. Finite Fields and Their Applications - Character Sums and Polynomials. De Grutyer, Radon Series on Computational and applied mathematics (11), 2013. [2] S. D. Cohen, S. Huczynska. The primitive normal basis theorem – without a computer. J. London Math. Soc. 67(1):41–56, 2003. [3] S. Huczynska, G. L. Mullen, D. Panario, D. Thomson. Existence and properties of k-normal elements over finite fields. Finite Fields Appl. 24:170–183, 2013. [4] G. Kapetanakis, L. Reis. Variations of the Primitive Normal Basis Theorem. Des. Codes Cryptgr. 87: 1459–1480, 2019. [5] H. W. Lenstra, Jr, R. J. Schoof. Primitive normal bases for finite fields. Math. Comp. 48(177):217–231, 1987. [6] H. Meyn. On the construction of irreducible self-reciprocal polynomials over finite fields. Appl. Algebr. Eng. Comm. 1(1): 43–53, 1990. [7] O. Ore. Contributions to the theory of finite fields. Trans. Amer. Math. Soc. 36 :243–274, 1934. [8] L. Reis. Existence results on k-normal elements over finite fields. Rev. Mat. Iberoam. 35(3): 805–822, 2019. [9] L. Reis, Contemporary topics in Finite Fields: Existence, characterization, construction and enumeration problems, PhD Thesis - Federal University of Minas Gerais, 2018. Link: http://hdl.handle.net/1843/EABA-B55PHT.

Departamento de Matematica,´ Universidade Federal de Minas Gerais, UFMG, Belo Horizonte MG (Brazil), 30123-970 E-mail address: [email protected]