Factoring Dickson polynomials over finite fields

Transcription

1 Factoring Dickson polynomials over finite fiels Manjul Bhargava Department of Mathematics, Princeton University. Princeton NJ Michael Zieve Department of Mathematics, University of Southern California. Los Angeles CA Abstract We erive the factorizations of the Dickson polynomials D n X, a) an E n X, a), an of the bivariate Dickson polynomials D n X, a) D n Y, a), over any finite fiel. Our proofs are significantly shorter an more elementary than those previously known. 1. Introuction. Let F q be the fiel containing q elements, an let p be the characteristic of F q. Let n be a nonnegative integer an a F q. The Dickson polynomial of the first kin, of egree n an parameter a, is efine to be the unique polynomial D n X, a) F q [X] for which D n Y + a/y ), a) = Y n + a/y ) n ; the Dickson polynomial of the secon kin, of egree n an parameter a, is efine to be the unique polynomial E n X, a) F q [X] for which E n Y + a/y ), a) = Y n+1 a/y ) n+1. The uniqueness of these Y a/y ) polynomials is clear; there are several ways to prove their existence, e.g. see [1] or [5, Lemma 1.1] or [4, 2.2)]. These polynomials have been extensively stuie, an in fact a book has been written in their honor [4]. In this note we shall erive the factorizations of the Dickson polynomials D n X, a) an E n X, a), an of the bivariate Dickson polynomial D n X, a) D n Y, a), over any finite fiel F q. In each case, our strategy will be to first write own the 1

2 factorization over the algebraic closure F q of F q, then etermine how to put together certain factors over F q in orer to get the irreucible factors over F q. The factorizations of D n X, a) an E n X, a) have been carrie out using much lengthier methos by W.-S. Chou [2]. The purpose of this note is to exhibit a simpler approach. Our methos can also be use to provie simple proofs of various other factorization results; as an example we inclue the factorization of D n X, a) D n Y, a) over any finite fiel, which seems to be new. Previously the factorization of this polynomial over the algebraic closure of a finite fiel was known, ue to K. S. Williams for n o [7] an to G. Turnwal for n even [5, Prop. 1.7]; we give simpler proofs of these results as well. We are grateful to G. Turnwal for informing us that our proof of this last result is similar to an argument of S. D. Cohen an R. W. Matthews [3, p. 67]. We shall retain the notation of the first paragraph throughout this note. Also, for any ξ F q, by ξ we shall mean a fixe square root of ξ in F q ; for any positive integer coprime to p, we will use ζ to enote a primitive -th root of unity in F q. 2. The Dickson polynomial of the first kin, D n X, a). Write n = p r m with p, m) = 1. Then it follows from the functional equation of D n that D n X, a) = D m X, a) pr ; thus, in orer to factor D n, it suffices to factor D m. Our first result gives the factorization of D m over F q. Theorem 1 For q o, D m X, a) = 2m 1 i o X a ζ i 4m + ζ i 4m) ) ; for q even, D m X, a) = X 2 m 1 X a ζ i m + ζ i m ) ) 2. Proof. Note that D m Y + a Y, a) = Y m + a/y ) m = ξ m = 1 a ) Y ξ ; Y 2

3 in orer to express the right-han sie as a function of Y + a/y ), we pair the terms corresponing to ξ an 1/ξ, which gives Y ξ a ) Y a ) = Y 2 ξ + 1 )a + Y a2 Y ξy ξ Y = + a ) 2 1 ) 2. a ξ + ξ 2 Y Thus, D m X, a) is the prouct of monic linear factors corresponing to the ξ F q for which ξ m = 1, where the factors corresponing to ξ an 1/ξ are X ± a ξ + 1/ ξ)). The result for q even follows at once; for q o, the result follows from the fact that the numbers ± ξ with ξ m = 1 are precisely the numbers ζ i 4m with i o an 0 < i < 4m. For a = 0, the factorization of D m X, a) = X m is trivial; our next two results give the factorization of D m X, a) over F q when a 0. Theorem 2 If q is o an a 0, then D m X, a) is the prouct of several istinct irreucible polynomials in F q [X], which occur in cliques corresponing to the ivisors of m for which m/ is o. To each such there correspon ϕ4)/2n ) irreucible factors, each of which has the form N 1 X a qi ζ qi 4 + ) ) ζ qi 4 for some choice of ζ 4 ; here ϕ enotes Euler s totient function, k is the least positive integer such that q k ±1 mo 4), an k /2 if a / F q an k 2 mo 4) an q k/2 2 ± 1 mo 4); N = 2k if a / F q an k is o; otherwise. k Proof. The previous result escribes the roots of D m X, a); clearly these roots are istinct, since ξ an ξ 1 are the only roots of the quaratic equation Z + Z 1 = ξ + ξ 1. Thus D m X, a) is the prouct of its istinct monic irreucible factors over F q, an each such factor is the minimal polynomial over F q of a root α of D m. These roots are given by α = a ζ 4 + ζ 1 4 ) where is a ivisor of m with m/ o, an ζ 4 is a primitive 4-th root of unity. The minimal polynomial of α over F q has the form N 1 X αqi ), where N enotes the least positive integer such that α qn = α. We will show 3

4 that N = N ; since for fixe there are ϕ4)/2 choices for α, the theorem follows. We now show N = N. Note that a ζ4 + ζ 1 4 )) q s = a qs ζ qs 4 + ) ζ qs 4 ; thus, N is the least positive integer s such that q s a ζ qs 4 + ) ) ζ qs 4 = a ζ4 + ζ 1 4. ) If a F q or s is even, then a qs = a, so ) just asserts that ζ qs 4 + ζ qs 4 = ζ 4 + ζ 1 4, or equivalently ζqs 4 = ζ±1 4, i.e. qs ±1 mo 4). If a / F q an s is o, then a qs = a, so ) is equivalent to ζ qs 4 = ζ±1 4, i.e. qs 2 ± 1 mo 4). The result follows at once by inspection. Theorem 3 If q is even an a 0, then D m X, a)/x is the prouct of the squares of several istinct irreucible polynomials in F q [X], which occur in cliques corresponing to the ivisors of m with > 1. To each such there correspon ϕ)/2k ) irreucible factors, each of which has the form k 1 X a ζ qi + ) ) ζ qi for some choice of ζ ; here k is the least positive integer such that q k ±1 mo ). Proof. By Theorem 1, the roots of the polynomial D n X, a)/x are the elements α = a ζ + ζ 1 ) where m an 1. As in the proof of Theorem 2, we conclue D m X, a)/x is the prouct of its istinct monic irreucible factors over F q, an each such factor is of the form N 1 X αqi ), where N enotes the egree of α over F q. Since q is even, a F q implies a Fq ; thus N is the least positive integer s such that ζ qs +ζ qs = ζ +ζ 1, i.e., q s = ±1 mo ). Hence N = k. Since for fixe there are ϕ)/2 choices for α, we have the theorem. 3. The Dickson polynomial of the secon kin, E n X, a). Write n+1 in the form p r m + 1), where p, m + 1) = 1. Using the functional equation for E n, we fin E n Y + a/y, a) = Y m+1 a/y ) m+1 ) pr Y a/y = E m Y + a/y, a) pr Y a/y ) pr 1, 4

5 an it follows that E n X, a) = E m X, a) pr X 2 4a) pr 1 2. Thus, to factor E n, it suffices to factor E m. Our first result gives the factorization of E m over F q ; we omit the proof since it is nearly ientical to that of Theorem 1. Theorem 4 For q o, for q even, E m X, a) = m X a ζ i 2m+1) + ζ i 2m+1) )) ; m/2 E m X, a) = X a ζ i m+1 + ζm+1) ) i 2. When a = 0, the factorization of E m X, a) = X m is trivial. In Theorems 5 an 6, we present the factorization of E m X, a) over the finite fiel F q in the case a 0. Theorem 5 If q is o an a 0, then E m X, a) is the prouct of several istinct irreucible polynomials in F q [X]. These occur in cliques corresponing to the ivisors of 2m + 1) with > 2. To each such there correspon ϕ)/2n ) irreucible factors, each of which has the form N 1 X a qi ζ qi + ) ) ζ qi for some choice of ζ, unless a is a nonsquare in F q an 4 ; in this exceptional case there are ϕ)/n factors corresponing to each of = 0 an = 2 0, where 0 > 1 is an o ivisor of m + 1, an the factors corresponing to 0 are ientical to the factors corresponing to 2 0. Here k is the least positive integer such that q k ±1 mo ), an k /2 if a / F q an 0 mo 2) an k 2 mo 4) an q k/2 ± 1 mo ); 2 N = 2k if a / F q an k is o; otherwise. k 5

6 We omit the proof since it is similar to that of Theorem 2. We remark that the corresponing result in [2], namely Thm. 3.1, is false in case a is a square in F q an 4 ; this case shoul be inclue in item 5) of that result rather than item 6). Theorem 6 If q is even an a 0, then E m X, a) is the prouct of the squares of several istinct irreucible polynomials in F q [X], which occur in cliques corresponing to the ivisors of m + 1 with > 1. To each such there correspon ϕ)/2k ) irreucible factors, each of which has the form k 1 X a ζ qi + ) ) ζ qi for some choice of ζ ; here k is the least positive integer such that q k ±1 mo ). Proof. When the characteristic is 2, we observe that E m Y + a/y, a) = Y m+1 a/y ) m+1 Y a/y = D m+1y + a/y, a) ; Y + a/y hence E m X, a) = D m+1 X, a)/x, so the esire factorization follows immeiately from Theorem The bivariate Dickson polynomial, D n X, a) D n Y, a). Write n = p r m, where m, p) = 1. Then the functional equation implies D n X, a) D n Y, a) = [D m X, a) D m Y, a)] pr ; thus, to factor D n X, a) D n Y, a), it suffices to factor D m X, a) D m Y, a). As the factorization of D m X, 0) D m Y, 0) = X m Y m is trivial, we shall assume a 0 throughout. Our first result gives the factorization of D m X, a) D m Y, a) over F q. Theorem 7 Let α i = ζ i m + ζ i m D m X, a) D m Y, a) = X Y ) an for m even, an β i = ζ i m ζ i m. Then for m o, m 1)/2 D m X, a) D m Y, a) = X Y )X + Y ) 6 X 2 α i XY + Y 2 + β 2 i a ), m 2)/2 X 2 α i XY + Y 2 + β 2 i a ).

7 Proof. Observe that a D m W + W, a) a D m Z + Z, a) = W m + a/w ) m Z m a/z) m [ a ) m ] = [W m Z m ] = ξ m =1 1 [ W Z W ) ξz 1 ξ a W Z )]. In orer to express the last expression as a function solely of W + a/w an Z + a/z, we pair the terms corresponing to ξ an 1/ξ; writing α = ξ + ξ 1 an β = ξ ξ 1, this gives = ) W ξz 1 ξ a W Z W + a ) 2 α W + a W W if ξ 1/ξ i.e. ξ ±1), an ) W ξz 1 ξ a ) = W Z ) W Z ) 1 a ξ ) Z + a ) + Z ) ξw Z Z + a ) 2 + β 2 a Z W + a ) ξ Z + a ) W Z otherwise. The factorization given in the theorem follows at once. Moreover, one can immeiately check that the given linear an quaratic factors are irreucible over F q ; hence the state factorization is complete. Our next result gives the factorization of D m X, a) D m Y, a) over F q. Theorem 8 The polynomial D m X, a) D m Y, a) is the prouct of istinct irreucible polynomials in F q [X], which occur in cliques corresponing to the ivisors of m. To each such 1, 2 there correspon ϕ)/2k ) irreucible factors of egree 2k, each of which has the form k 1 X 2 α qi XY + Y 2 + β 2qi a). For {1, 2}, there correspons a single factor of the form X ζ Y ). Here α an β enote ζ + ζ 1 an ζ ζ 1 respectively for some choice of ζ, an k is the least positive integer such that q k ±1 mo ). 7

8 Proof. Note that the quaratic factors of D m X, a) D m Y, a) over F q as given in the previous theorem are istinct, since α 1,..., α m 1)/2 are istinct. Hence D m X, a) D m Y, a) is the prouct of its istinct monic irreucible factors over F q, an each such factor is either of the form X ζ Y ) for = 1 or 2, or is of the form N 1 X 2 α qi XY + Y 2 + β 2qi a), for some m with > 2, where N enotes the least positive integer such that both α an β 2 are elements of F q N. However, note that β2 = α2 4 F q α ), an, as before, the smallest integer M such that α qm = α is M = k. Hence N = k, an the theorem follows. Remark. There oes not appear to be an analogous way to treat the bivariate Dickson polynomial E n X, a) E n Y, a) of the secon kin, an in fact little is known about this factorization, although G. Turnwal has some preliminary results [6]. Finally, we mention one further factorization involving Dickson polynomials: q 1 E i X, 1)Y q 1)q 1 i) = a F q [D q 1 Y, a) X]. Many further results along these lines can be foun in [1]. Acknowlegments. That the arguments presente in [2] can be greatly simplifie has been inepenently notice by S. Gao private communication). The first author acknowleges support from AT&T Research uring the summer of The secon author was supporte by an NSF Postoctoral Fellowship. References [1] S. S. Abhyankar, S. D. Cohen, an M. E. Zieve, Bivariate factorizations connecting Dickson polynomials an Galois theory, Trans. Amer. Math. Soc., to appear. [2] W. Chou, The factorization of Dickson polynomials over finite fiels, Finite Fiels Appl ),

9 [3] S. D. Cohen an R. W. Matthews, Monoromy groups of classical families over finite fiels, in Finite Fiels an Applications, LMS Lecture Note Series, Vol. 233, pp , Cambrige, [4] R. Lil, G. L. Mullen, an G. Turnwal, Dickson Polynomials, Pitman Monographs an Surveys in Pure an Applie Math.,Longman, Lonon/Harlow/Essex, [5] G. Turnwal, On Schur s conjecture, J. Austral. Math. Soc. Ser. A) ), [6] G. Turnwal, Monoromy groups of Dickson polynomials of the secon kin, preprint, [7] K. S. Williams, Note on Dickson s permutation polynomials, Duke Math. J ),