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ON CERTAIN CHARACTER SUMS OVER Fq[T] 1. Introduction In PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 126, Number 3, March 1998, Pages 647{652 S 0002-9939(98)04582-1 ON CERTAIN CHARACTER SUMS OVER Fq[T ] CHIH-NUNG HSU (Communicated by Dennis A. Hejhal) Abstract. Let Fq be the finite field with q elements and let A denote the ring of polynomials in one variable with coefficients in Fq .LetPbe a monic polynomial irreducible in A. We obtain a bound for the least degree of a monic polynomial irreducible in A (q odd) which is a quadratic non-residue modulo P . We also find a bound for the least degree of a monic polynomial irreducible in A which is a primitive root modulo P . 1. Introduction In [1], on the assumption of the Extended Riemann Hypothesis, Ankeny proved that the least positive quadratic non-residue of the prime k is O((log k)2)andthe ν(k 1) ν(k 1) 2 least positive primitive root (mod k)isO (2 − log k(log 2 − log k)) ,where ν(k 1) denotes the number of distinct prime{ factors of k 1. } − − Let Fq be the finite field with q elements and let A denote the ring of polynomials in one variable with coefficients in Fq.LetPbe a monic irreducible in A.Inthis note, we establish the following results: (1) When q is odd, the least degree of a monic irreducible in A which is a quadratic non-residue modulo P is less than 2 + 2 logq(1 + deg P ) (corollary 2:2). In fact, this result is deduced from a more general situation (proposition 2:1). (2) The least degree of a monic irreducible in A which is a primitive root modulo deg P qdeg P 1 P is O( log deg P ) (theorem 3:1). Moreover, if q 1− is a prime number, then q − the least degree of a monic irreducible primitive root modulo P is less than 8+2logqdeg P (proposition 3:1). The above results will be deduced from a character sum estimate (theorem 2:1) due to Effinger and Hayes [3], Chapter 5. 2. The least positive quadratic non-residues Let Fq denote the finite field with q elements and let A denote the ring of polyno- mials in one variable T with coefficients in Fq.WedenotebyA+the set consisting (n) of all positive (monic) polynomials in A,andbyA+ the set consisting of all pos- itive polynomials in A+ of degree n. We may write any element f of A in the Received by the editors August 20, 1996. 1991 Mathematics Subject Classification. Primary 11A07; Secondary 11L40, 11N05. Key words and phrases. Riemann Hypothesis, quadratic non-residues, primitive roots. c 1998 American Mathematical Society 647 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 648 CHIH-NUNG HSU form d i f = aiT with ai Fq and ad =0: ∈ 6 i=0 X The degree of f is defined by deg f = d, and the valuation of f is defined by f = qd. | | It is known that the number πn of all positive irreducibles in A+ of degree n satisfies n n q n q (1) q 2 +1 π : n − ≤ n ≤ n Suppose that ∆ is a positive polynomial in A+. A character χ modulo ∆ is a group homomorphism χ :(A=∆)× C×. We define χ(f)=0if(f;∆) = 1 and define → 6 πn(χ)by πn(χ)= χ(P ): irreducible P A(n) X ∈ + These characters modulo ∆ can be identified as the idele class characters of Dirichlet type for the rational function field Fq(T ) (cf. [3], Exercise 2 of Section 5.1). Theorem 2.1. Let χ be a non-trivial character modulo ∆.Then n q 2 π(χ) (deg ∆ + 1) : | n |≤ · n Proof. Let m(χ) be the conductor of the idele class character χ. It follows from [3], Exercise 2 of Section 5.1, that m(χ) ∆ifχis ramified at and m(χ) ∆ifχ is unramified at . From [3], theorem|∞ 5.7, we know that ∞ | ∞ n q2 π (χ) deg m(χ) λ(χ)+2 ; | n |≤{ − }· n where λ(χ)=2ifχis ramified at and λ(χ)=1ifχis unramified at .This completes the proof. ∞ ∞ Theorem 2.2. Let χ be a non-trivial character modulo ∆ and let the complex number ξ with ξ =1be a value of the character χ . If a positive integer n satisfies | | 1+2log 1+deg ∆ for 5 q √q 2 q ; − ≥ n 2+2log (1+deg∆) for q =3;4; ≥ 8 q > <5+2logq(1+deg∆) for q =2; > (n) there then exists at least: one positive irreducible P in A+ such that (P; ∆) = 1 and χ(P ) = ξ. 6 Proof. Suppose that χ(P )=ξfor all positive irreducible P in A with deg P = n and (P; ∆) = 1. By the assumption, we have π deg ∆ π (χ): n − ≤| n | Using (1), we get n q n q 2 deg ∆ <π deg ∆; n − − n− License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use ON CERTAIN CHARACTER SUMS OVER Fq[T ] 649 and by theorem 2:1, we obtain n q 2 π (χ) (deg ∆ + 1) : | n |≤ · n Combining these, we obtain (under the above assumption) n n q n q 2 (2) q 2 deg ∆ < (deg ∆ + 1) : n − − · n If 1+2log 1+deg ∆ and 5, then we have n q √q 2 q ≥ − ≥ 1+deg∆ √q 2 n 1 −2 n − 1;q n: q2 ≤ √q ≤ ≥ This implies that n n 1 n 1 q 2 deg ∆ (√q 2) q −2 q −2 1 n − · + 2 2 n − − q 2 ≥ n · n − ! 1+deg∆ 1 +0 (deg ∆ + 1) : ≥ n ≥ · n Thus we obtain n n q n q 2 q 2 deg ∆ (deg ∆ + 1) : n − − ≥ · n This contradicts (2); hence it contradicts the assumption. There thus exists at least (n) one positive irreducible P in A+ such that (P; ∆) = 1 and χ(P ) = ξ. The proofs of the other cases are similar. 6 Corollary 2.1. Let χ be a non-trivial character modulo ∆.Ifq 5and deg ∆ ≥ (2) ≤ q 2√q 1, then there exists at least one positive irreducible P in A+ such that χ(−P ) =1−. 6 Proof. This follows immediately from theorem 2:2. If ∆ = P is a positive irreducible in A+, we then have the following: Proposition 2.1. Let χ be a non-trivial character modulo P and let the complex number ξ with ξ =1be a value of the character χ. If a positive integer n satisfies | | 1+2log 1+deg P for 4 q √q 1 q ; − ≥ (3) n 2+2log (1+degP) for q =3; ≥ 8 q > <4+2logq(1+degP) for q =2; > (n) there then exists at least: one positive irreducible P 0 in A such that P 0 = P and + 6 χ(P 0) = ξ. 6 Proof. The proof is a modification of the proof of theorem 2:2. Corollary 2.2. Let P be a positive irreducible in A+ (q odd). If n satisfies the condition (3), then there exists at least one positive irreducible in A+ of degree n which is a quadratic non-residue modulo P and at least one positive irreducible in A+ of degree n which is a quadratic residue modulo P . License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 650 CHIH-NUNG HSU Proof. In proposition 2:1, let χ be the quadratic symbol for A (cf. [2]). The corollary then follows immediately. Corollary 2.3. Let P be a positive irreducible in A+, and let the positive integer d divide P 1.Ifnsatisfies the condition (3), there then exists at least one positive | |− irreducible in A+ of degree n which is a d-th power non-residue modulo P . Proof. Since the unit group (A=P A)× is a cyclic group, its character group is cyclic of order P 1. This corollary then follows immediately from d P 1and proposition 2:1.| |− || |− Let norm be the norm homomorphism of (A=P A)× onto Fq×,wherePis a positive irreducible in A+.Letχ:Fq× C×be any non-trivial character. Then, by proposition 2:1, we also have → Corollary 2.4. If n satisfies the condition (3), there then exists at least one posi- tive irreducible P 0 in A+ of degree n such that χ norm(P 0) =1. ◦ 6 3. The least positive primitive roots Let P be a positive irreducible in A+ with deg P 2. We say that a polynomial ¯ ≥ P 0 is a primitive root modulo P if P 0 is a generator of the cyclic group (A=P A)×, ¯ where P 0 is the canonical image of P 0 in A=P A. The purpose of this section is to find a bound for the least degree of a positive irreducible primitive root modulo P . Theorem 3.1. There exists a positive number c such that for any positive irre- deg P ducible P in A+,ifn c log deg , one can find at least one positive irreducible ≥ · q P of degree n which is a primitive root modulo P . Proof. Since (A=P A)× is a cyclic group of order P 1, the group CP consisting of all characters modulo P is also a cyclic group of| order|− P 1. If m is a positive integer such that m ( P 1), then| it|− is known that for any | | |− P 0 A ∈ P 1 | |− m if P 0 m 1(modP); χ(P0)= ≡ m 0otherwise; χ C ,χ =χ0 ( ∈ PX where χ0 is the trivial character modulo P .
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