On the Reducblty of Cyclotomc Polynomals over Fnte Felds Brett Harrson under the drecton of Mk Havlíčková Massachusetts Insttute of Technology Research Scence Insttute August 2, 2005
Abstract The rreducblty of a polynomal n a prme modulus mples rreducblty over the ratonals. However, the converse s certanly not true. In fact, there are some polynomals that are rreducble over the ratonals yet reducble n every prme modulus. We show that the nth cyclotomc polynomal reduces modulo all prmes f and only f the dscrmnant of the nth cyclotomc polynomal s a square. We pose further questons about the specfc factorzaton of cyclotomc polynomals over fnte felds n relaton to the dscrmnant.
1 Introducton A polynomal f Q[x] that s rreducble over Q may stll factor nto smaller degree polynomals over Z/pZ for some prme p. For example, let f(x) =x 4 + 1. It can easly be shown that f s rreducble over Q usng Esensten s Crteron [7, pgs. 23-24]. Yet over Z/2Z, f s reducble, snce x 4 +1 (x 2 +1)(x 2 +1) (mod 2). In fact, t can be shown that f s reducble n every prme modulus. In general, t s dffcult to predct when a partcular polynomal wll have ths property. As the man result of ths paper, we prove the followng theorem whch provdes a method for determnng exactly when the nth cyclotomc polynomal s reducble n all prme modul: Theorem 1. For n 3, the nth cyclotomc polynomal, Φ n (x), s reducble modulo all prmes f and only f the dscrmnant of Φ n (x) sasquarenz. In secton 2, we provde several defntons and termnology relatng to cyclotomc polynomals, Galos theory, the dscrmnant, and cyclotomc feld extensons. In secton 3, we ntroduce several lemmata necessary for the proof of the man theorem whch we carry out n secton 4. In secton 5, we conclude our results and offer further drectons for research. 2 Defntons and Termnology 2.1 Cyclotomc Polynomals Defnton. Let ζ n denote a prmtve nth root of unty. The mnmal polynomal (see Appendx A) of ζ n over Q s called the nth cyclotomc polynomal, denoted by Φ n (x), and can be defned as the product Φ n (x) = 0<<n gcd(,n)=1 ( x ζ n ). 1
It follows that the degree of Φ n (x) sϕ(n), where ϕ(n) denotes the Euler ϕ-functon (see Appendx D). It s well-known that Φ n (x) s rreducble over Q [3, pgs. 234 235]. 2.2 Galos Correspondence For a feld K, let α 1,...,α n be the roots of an rreducble polynomal f K[x]. Let L = K(α 1,...,α n ), the feld extenson of K obtaned by adjonng the elements α 1,...,α n (see Appendx A). Defnton. The set of automorphsms of L that fx the elements of K form a group by composton (see Appendx B). Ths group s called the Galos group of the feld extenson L K and s denoted by Γ ( L K ). The elements of Γ ( L K ) permute the roots of f, snce the automorphsms must preserve the coeffcents of f over K. Thus,Γ ( L K ) s a subgroup of S n, the symmetrc group on n elements (see Appendx C). If H Γ ( L K ) s a subgroup of Γ ( L K ),thenh fxes some ntermedate feld of L K. Clearly, the dentty fxes all the elements of L whle the entre Galos group Γ ( L K ) fxes only K. Ths establshes an order-reversng, one-to-one correspondence between the subgroups of Γ ( L K ) and the ntermedate felds of L K, n accordance wth the Fundamental Theorem of Galos Theory [7, pgs. 114-117]. 2.3 Cyclotomc Extensons and the Dscrmnant Recall from secton 2.1 that the degree of Φ n (x) sϕ(n) andthatφ n (x) s rreducble over Q. It follows that [Q(ζ n ):Q] =ϕ(n), where [Q(ζ n ):Q] denotes the degree of the extenson Q(ζ n ) Q (see Appendx A). Further, Γ ( Q(ζ n ) Q ) s somorphc to (Z/nZ),where(Z/nZ) denotes the multplcatve group modulo n [3, pg.235]. 2
Defnton. Let f Q[x] be defned as f(x) =(x α 1 ) (x α n ). The dscrmnant of f, denoted ( f ), s defned as the product ( f ) = <j(α α j ) 2. An explct formula for ( Φ n (x) ) can be obtaned. Let s be the number of unque prme dvsors of n. Then, ( Φ n (x) ) can be expressed as [6, pg.269] ( Φ n (x) ) =( 1) (1/2)ϕ(n)s q n q prme n ϕ(n). (1) ϕ(n)/(q 1) q Defnton. For a feld extenson L K, thefeld dscrmnant of L, denoted δ L, s the dscrmnant of the mnmal polynomal of L K. By defnton, t drectly follows that δ Q(ζn) = ( Φ n (x) ). Throughout the paper, these two notons of dscrmnant wll be used nterchangeably. 3 Lemmata The followng well-known lemmata wll be used n the development of the man theorem. Lemma 3.1. For a Z, the equaton x 2 = a has a soluton n Z f and only f x 2 a (mod n) has a soluton for all n Z. Lemma 3.2. Let F p denote the fnte feld of p elements, Z/pZ. For all feld extensons K F p, Γ ( K F p ) s somorphc to Ck for some k, wherec k denotes the cyclc group on k elements (see Appendx C). Lemma 3.3. [3, pgs. 168-169] For a feld extenson L K, Γ ( L K ) A n f and only f δ L s asquarenz, wherea n denotes the alternatng group on n elements (see Appendx C). 3
Lemma 3.4. For n 3, (Z/nZ) has a generator f and only f n =4, q k,or2q k for any odd prme q and postve nteger k. Lemma 3.5 (Frobenus Densty Theorem). [2] Let f be an rreducble polynomal of degree n over Q, letn be the splttng feld of f (see Appendx A), andletg =Γ ( N Q ). If G contans a permutaton σ whch s the product of dsjont cycles of length n 1,...,n k, then there exsts an nfnte set P σ of prmes such that for any p P σ we have the followng decomposton of f over F p : k f(x) = f (x), where all f are rreducble over F p and the degree of f s n. 4 Proof of the Man Theorem Recall the example from secton 1, f(x) =x 4 + 1, whch s the 8th cyclotomc polynomal Φ 8 (x). Computaton shows that ( Φ 8 (x) ) = 256 = 16 2. If one computes the dscrmnants for the frst several cyclotomc polynomals that reduce modulo all prmes, one fnds that they are all squares. These observatons motvate the man theorem: Theorem 1. For n 3, Φ n (x) s reducble modulo all prmes f and only f ( Φ n (x) ) s a square n Z. We begn by provng that f ( Φ n (x) ) s a square n Z, thenφ n (x) s reducble modulo all prmes. Suppose ( Φ n (x) ) s a square n Z. By Lemma 3.1 and Lemma 3.3, we know that Γ ( ) F p (ζ n ) F p Aϕ(n). By Lemma 3.2, we also know that Γ ( ) F p (ζ n ) F p = Ck for some k. The elements of the Galos group permute the ϕ(n) rootsofφ n (x), and so k ϕ(n). But ϕ(n) sevenforn 3, and k cannot be even snce the sgn of any cycle of even length s 1 (see Appendx C). Therefore, ( ) Γ Fp (ζ n ) F p <ϕ(n) and the mnmal polynomal of 4
F p (ζ n ) F p has degree less than the degree of Φ n (x). Ths mples that Φ n (x) reduces over F p for all p. Proceedng n the other drecton, we show that f ( Φ n (x) ) s not a square n Z, then Φ n (x) s rreducble over F p for some prme p. Frst assume (Z/nZ) has a generator g. Then over Q, the automorphsm ζ n ζn g generates a cycle of length ϕ(n), and therefore Γ ( Q(ζ n ) Q ) s somorphc to C ϕ(n). By Lemma 3.5, snce C ϕ(n) s the product of one dsjont cycle of length ϕ(n), there are nfntely prmes p for whch Φ n (x) s rreducble over F p and we are done. We must now show that f ( Φ n (x) ) s not a square n Z, then(z/nz) has a generator. Or equvalently, f (Z/nZ) does not have a generator, then ( Φ n (x) ) s a square n Z. Lemma 3.4 reduces the proof to showng that f n 4,p k,or2p k for any odd prme p and postve nteger k, then ( Φ n (x) ) s a square n Z. We wll analyze the possble cases for n 4,p k,or2p k to prove the theorem. Two more lemmata are requred. Lemma 4.1. Let z = p e 1 1 p e k k,wherethep are dstnct odd prmes. Then, ( Φ z (x) ) = ( Φ 2z (x) ). Proof. Snce ϕ(n) s multplcatve, we have that ϕ(2z) = ϕ(2)ϕ(z) = ϕ(z). From (1), ( Φ 2z (x) ) = 2 ϕ(z) k (2z) ϕ(z) p ϕ(z)/(p 1) = 2 ϕ(z) k 2 ϕ(z) z ϕ(z) p ϕ(z)/(p 1) = ( Φ z (x) ). Lemma 4.2. Let z =2 e 0 p e 1 1 pe k k,wherethep are dstnct odd prmes. Then, ( Φ z (x) ) = k ( ) p ϕ(z) e p e 1 /(p 1). (2) 5
Proof. ( Φ z (x) ) = = ( k p k z ϕ(z) p ϕ(z)/(p 1) = k e ϕ(z)(p 1) ϕ(z) /(p 1) = ) k p e ϕ(z) k p ϕ(z)/(p 1) ( ) p ϕ(z) e p e 1 /(p 1). Wth these two lemmata, the three cases for whch n 4,p k,or2p k can be examned: Case 1: n = p e 1 1 p e k k where the p are dstnct odd prmes and k 2. Usng equaton (2), the exponent of p n ( Φ n (x) ) s ϕ(z) ( e p e 1 ) /(p 1). Snce p 1 dvdes p e p e 1, t follows that p 1 dvdes ϕ(z). Moreover, ϕ(z)/(p 1) s even snce z s the product of at least two dstnct odd prmes (see Appendx D). Thus, ( Φ n (x) ) = k ( ) p ϕ(z) e p e 1 /(p 1) = k ( ) p (1/2)ϕ(z) e p e 1 /(p 1), whch s an nteger. From equaton (2), the sgn of ( Φ n (x) ) s ( 1) (1/2)ϕ(n)s whch equals 1 snce 4 dvdes ϕ(n) fn s the product of two or more dstnct odd prmes. Therefore, ( Φ n (x) ) Z. Case 2: n =2p e 1 1 p e k k,wherethep are dstnct odd prmes and k 2. Ths s analogous to Case 4 by Lemma 4.1. Case 3: n =2 e 0 p e 1 1 pe k k,wherethep are odd prmes, e 0 2, andk 1. Snce ϕ(2 e 0 ) s even, ϕ(z)/(p 1) s once agan even and the case becomes analogous to Case 4. So f n 4,p k,or2p k,then ( Φ n (x) ) s a square n Z, and the proof s complete. 6
5 Concludng Remarks By provng Theorem 1, we have provded an easy method for determnng when the nth cyclotomc polynomal s reducble modulo all prmes. In addton, we hghlght an nterestng corollary that results from the proof of Theorem 1. In the second part of the proof, we showed that f ( Φ n (x) ) s not a square n Z, then(z/nz) has a generator, whch mples that Γ ( Q(ζ n ) Q ) s somorphc to C ϕ(n). Usng Lemma 3.3, we note the followng corollary: Corollary 5.1. If Γ ( Q(ζ n ) Q ) A ϕ(n),thenγ ( Q(ζ n ) Q ) = Cϕ(n). Whle Theorem 1 descrbes when a cyclotomc polynomal s reducble modulo all prmes, t stll remans open exactly how these polynomals reduce for partcular modul. We hope to fnd better methods of predctng exactly how a cyclotomc polynomal splts over dfferent fnte felds. It s also unsolved whether the Galos groups of cyclotomc extensons are further affected by dscrmnants that are hgher powers of ntegers. We may fnd that the dscrmnant of a cyclotomc polynomal can tell us exactly how the polynomal factors over a fnte feld. 6 Acknowledgments I would lke to thank Mk Havlíčková, of the mathematcs department of the Massachusetts Insttute of Technology, for teachng, gudng, and supportng me throughout my research. I would also lke to thank Dr. John Rckert for all of hs assstance wth my project and Jen Balakrshnan for her helpful comments. Fnally, I would lke to thank the Center for Excellence and Educaton and the Research Scence Insttute for provdng me wth the opportunty and the resources to conduct my research. 7
References [1] M. Artn, Algebra. Prentce-Hall, Upper Saddle Rver, NJ, 1991. [2] R. Brandl, Integer Polynomals that are Reducble Modulo all Prmes. The Amercan Mathematcs Monthly, Vol. 93, No. 4 (Apr., 1986), 286-288. [3] D.A.Cox,Galos Theory. John Wley & Sons, Hoboken, NJ, 2004. [4] K. Ireland and M. Rosen, A Classcal Introducton to Modern Number Theory. Sprnger- Verlag, New York, NY, 1990. [5] I. Nven and H. S. Zuckerman, An Introducton to the Theory of Numbers. John Wley & Sons, New York, NY, 1991. [6] P. Rbenbom, Algebrac Numbers. John Wley & Sons, New York, NY, 1972. [7] I. Stewart, Galos Theory. Chapman and Hall, London, 1973. 8
Appendx A: Groups and Felds A group s a set G together wth a law of composton whch s assocatve and has an dentty element, and such that every element of G has an nverse. An abelan group s a group whose law of composton s commutatve. Examples of groups nclude S n, C n,andz(see Appendx C). Note that Z and C n are abelan groups, whle S n for n 3snot.Theorder of a group G, denoted G, s the number of elements of G. G may be nfnte. A subset H of a group G s called a subgroup f t has the followng propertes: 1. Closure: If a H and b H, thenab H. 2. Identty: 1 H. 3. Inverses: If a H, thena 1 H. For a subgroup H of a group G, aleft coset of G s a subset of G of the form: ah = {ah h H}. Smlarly, a rght coset of G s a subset of G of the form Ha. H s called a normal subgroup of G f for a G, aha 1 H for all h H. A subgroup s normal f and only f ts left cosets are the same as ts rght cosets, that s, f and only f ah = Ha for all a G. A feld F s a set together wth two laws of composton, addton and multplcaton, that satsfy the followng axoms: 1. Addton makes F nto an abelan group. Its dentty element s denoted by 0. 2. Multplcaton s assocatve and commutatve and makes the nonzero elements of F nto a group. Its dentty element s denoted by 1. 3. Dstrbutve law: For all a, b, c F,(a + b)c = ac + bc. 9
Q, R, andc are all examples of felds. A fnte feld s a feld contanng fntely many elements. A subfeld L of a feld F s a subset of F that s closed under addton, subtracton, multplcaton, and dvson, and that contans the dentty elements of F. The characterstc of a feld s the smallest postve nteger n for whch 1 } + +1 {{} = 0. If ths sum s never 0, n tmes the feld s sad to have characterstc 0. A feld extenson K of a feld F, denoted K F, s a feld contanng F as a subfeld. For a feld F and elements α 1,...,α n,letf (α 1,...,α n ) denote the smallest extenson of F contanng α 1,...,α n. A feld extenson F (α 1,...,α n ) s called algebrac f α 1,...,α n are all of roots of a polynomal over F.Themnmal polynomal of F (α 1,...,α n ) F s the unque monc nonconstant polynomal f F [x] satsfyng the followng two propertes [3, pg. 74]: 1. f(α )=0for1 n. 2. If g F [x] sanypolynomalwthα as a root, then g s a multple of f. The degree of a feld extenson K F, denoted [K :F ], s the dmenson of K as a vector space over F. For an extenson F (α) F where α s the root of an rreducble polynomal f F [x], [K :F ] s equal to the degree of f. IfF K L are felds, then [L:F ]=[L:K][K :F ] [1, pgs. 497-498]. Let f be an rreducble polynomal over a feld F.Thesplttng feld of f s the smallest extenson of F over whch f splts completely nto lnear factors. Appendx B: Morphsms Let G, H be groups. A homomorphsm φ: G H s any map satsfyng: φ (ab) =φ (a) φ (b), for all a, b G. Themage of a homomorphsm φ, denoted m φ, sthesetofally H for whch y = φ (x) forsomex G. Thekernel of φ, denoted ker φ, sthesetofallx G for 10
whch φ (x) = 1. It s clear that ker φ s a subgroup of G and m φ s a subgroup of H. Amapφs surjectve f m φ = H, that s, f every y H has the form φ (x) forsome x G. Amapφ s njectve f, for two elements x 1,x 2 G, x 1 x 2 mples φ (x 1 ) φ (x 2 ). Fnally, a map s bjectve f t s both surjectve and njectve. A bjectve homomorphsm φ: G H s called an somorphsm, andg s sad to be somorphc to H, denoted G = H. An somorphsm G G s called an automorphsm. Appendx C: Examples of Fnte Groups C n denotes the cyclc group of n elements, generated by an element x satsfyng x n =1. Thus, C n = {1,x,...,x n 1 },whereforall0, j < n, j mples x x j. S n denotes the symmetrc group of n elements. It s the group of permutatons of the set of ntegers from 1 to n, and thus S n = n!. For example, S 3 s the group of permutatons on the ntegers 1, 2, 3. Let 1 denote the dentty and let (a 1...a k )denotethemapa 1 a k a 1. Then, S 3 = {1, (1 2), (2 3), (1 3), (123), (1 3 2)}. A sngle permutaton of the form (a a j ) s called a transposton. The sgn functon of a permutaton σ, denoted sgn (σ), s ether equal to 1 or 1, dependng on whether σ s the composton of an odd or even number of transpostons, respectvely. For example, sgn (2 3) = 1 whle sgn (1 2 3) = 1, snce (1 2 3) = (1 2)(2 3). It follows that the sgn of any cycle of even length s 1 whle the sgn of any cycle of odd length s 1. The sgn functon s well defned; that s, f a permutaton σ can be expressed as the composton of two dfferent numbers of transpostons j and k, thenj k (mod 2) and so ther sgns are the same. A n denotes the alternatng group of n elements, consstng of the set of all even permu- 11
tatons of n elements. That s: A n = {σ S n sgn (σ) =1}. It follows that A n s a subgroup of S n,and A n = 1 2 n!. Appendx D: The Euler ϕ-functon The Euler ϕ-functon s an arthmetc functon defned by the followng: ϕ(n) = {a 1 a n, gcd(a, n) =1}. It follows that for a prme p, ϕ(p) =p 1 and more generally, ϕ(p k )=p k p k 1. The ϕ-functon s multplcatve, that s, for coprme ntegers a and b, ϕ(ab) = ϕ(a)ϕ(b). Thus, a general formula for ϕ(n) can be obtaned. Let n = p e 1 1 p e k k where the p are prmes. ϕ(n) = k ( p e ) p e 1. If p s odd, then each factor p e p e 1 s even and so 2 dvdes p e f n s the product of two or more dstnct prme factors. p e 1.Thus,4 ϕ(n) 12