Factoring polynomials over Z4 and over certain Galois rings

Transcription

1 Loughborough Unversty Insttutonal Repostory Factorng polynomals over Z4 and over certan Galos rngs Ths tem was submtted to Loughborough Unversty's Insttutonal Repostory by the/an author. Ctaton: SALAGEAN, A.M., Factorng polynomals over Z4 and over certan Galos rngs. Fnte felds and ther applcatons, 11 (1), pp Addtonal Informaton: Ths artcle was publshed n the journal, Fnte felds and ther applcatons [ c Elsever] and s also avalable at: Metadata Record: Publsher: c Elsever Please cte the publshed verson.

2 Ths tem was submtted to Loughborough s Insttutonal Repostory by the author and s made avalable under the followng Creatve Commons Lcence condtons. For the full text of ths lcence, please go to:

3 Factorng polynomals over Z 4 and over certan Galos rngs Ana Sălăgean Department of Computer Scence Loughborough Unversty, UK A.M.Salagean@lboro.ac.uk May 7, 2004 Abstract It s known that unvarate polynomals over fnte local rngs factor unquely nto prmary parwse coprme factors. Prmary polynomals are not necessarly rreducble. Here we descrbe a factorsaton nto rreducble factors for prmary polynomals over Z 4 and more generally over Galos rngs of characterstc p 2. An algorthm s also gven. As an applcaton, we factor x n 1 and x n + 1 over such rngs. Keywords: Polynomal factorng, Galos rngs, cyclc codes over rngs 1 Introducton Unvarate polynomals over a fnte local rng factor unquely nto prmary parwse coprme factors (see [9]). A prmary polynomal mght be rreducble (for example x s rreducble n Z 4 [x]) or reducble, n whch case ts factorsaton wll n general not be unque (for example x 2 = (x + 2) 2 n Z 4 [x]). Not even the number of factors and ther degrees are unque (for example x 4 = (x 2 + 2) 2 n Z 4 [x]). We descrbe a factorsaton of prmary polynomals nto rreducble factors over a Galos rng of characterstc p 2 (p beng a prme), gvng also an algorthm. The factorsaton we obtan has the property that t has the maxmum number of rreducble factors; moreover, among all factorsaton nto the maxmum number of rreducble factors, t has the mnmal number of dstnct factors (ths number wll turn out to be always one or two). We also descrbe all the factorsatons nto the maxmum number of rreducble factors. Our nterest n polynomals over Z 4, and more generally, Galos rngs was motvated by the exstence of good error-correctng codes over Z 4 and over Galos rngs [8]. Cyclc codes of length n over a rng R are deals n R[x]/ x n 1. So the factorsaton of x n 1 s partcularly mportant for ths applcaton. Another closely related motvaton comes from sequences over Z 4 and over Galos rngs. Here agan polynomals of the form x n 1 play an mportant role. As all recurrent sequences are perodc, they are n partcular lnearly recurrent and satsfy the lnear recurrence (of not necessarly mnmal degree) defned by x n 1, wth n the perod of the sequence. An algorthm for determnng all factorsatons of a polynomal over a rng of the form Z p a (and some other types of rngs) was developed n [13]. One factorsaton s derved from the factorsaton of the polynomal over the p-adc ntegers (ths can be obtaned by the algorthms of Chstov, Ford-Zassenhaus, Buchmann-Lenstra, Cantor-Gordon, Paul, Ford et. al., see [2, 1

4 4, 5, 6, 7, 10] ). However, ths approach only works when the dscrmnant of the polynomal (as a p-adc number) s not a multple of p a. (For example, t cannot be drectly appled to factorng x n 1 over Z 4 when n s even.) Factorng over the p-adcs and then projectng the factorsaton to Z p a[x] does not always result n a factorsaton nto rreducble factors, as rreducble monc polynomals over the p-adc ntegers may no longer be rreducble when projected (see Example 4.6 for llustraton). The advantage of our results compared to [13] s that they hold for all polynomals, regardless of the value of ther dscrmnant. The dsadvantage s that they only hold n Galos rngs of characterstc p 2, wth no mmedate way of extendng them to Galos rngs of characterstc p a wth a > 2. The paper s organsed as follows. We start by recallng known results n Secton 2. Secton 3 gves an rreducblty crteron for polynomals over a Galos rng. We then restrct our attenton to Galos rngs of characterstc p 2 and fully descrbe n Secton 4 factorsatons of the prmary polynomals n ths case. An algorthm wll also result. We also note an nterestng connecton between the factorsaton of a polynomal f and GR(p 2, r)[x]/ f beng a prncpal deal rng (see Theorem 4.10). In Secton 5 we apply our results to factorng x n 1 and x n + 1 over Galos rngs of characterstc p 2 (ncludng Z 4 as an mportant specal case). 2 Prelmnares Recall that f K s a feld, K[x] s a unque factorsaton doman. A polynomal s prme f and only f t s rreducble. When K s a fnte feld there are algorthms for factorng a polynomal nto rreducble factors over K[x] (see [1]). We wll recall some known results on the factorsaton of polynomals over a fnte local rng, followng manly [9]. Let R be a fnte local rng and let M be ts maxmal deal. All elements of M are nlpotent and all elements of R \ M are unts. The feld K := R/M s called the resdue feld of R. We denote by c the mage of c R under the canoncal projecton from R to K. Ths projecton extends naturally to a projecton from R[x] to K[x]. We wll call a polynomal monc f ts leadng coeffcent s 1. A polynomal n R[x] s called regular f t s not a zero-dvsor. Theorem 2.1 ([9, Theorems XIII.2 and XIII.6]) Let f = m =0 c x R[x]\{0}. Then: () f s a zero-dvsor ff c M for = 0,..., m; () f s a unt ff c 0 s a unt and c M for = 1,..., m; () f s regular ff there s an, 0 m such that c s a unt; (v) If f s regular then there are unque polynomals f, u R[x] such that f = uf, u s a unt and f s monc. So based on Theorem 2.1(v) we can assume that a regular polynomal s monc. Also, when lookng at factorsatons of a monc polynomal we can assume, wthout loss of generalty, that all factors are monc. Prme polynomals are rreducble. However, unlke n the case of felds, rreducble polynomals need not be prme. Recall that a polynomal f R[x] s called basc rreducble f f s rreducble n K[x]. Obvously, basc rreducble polynomals are rreducble. A polynomal f R[x] s called prmary f the deal f s prmary n R[x],.e. f for all gh f we have g f or h m f for some nteger m 1. Prmary polynomals n K[x] are powers of prme polynomals. Prmary polynomals n R[x] are charactersed below: 2

5 Theorem 2.2 ([9, Proposton XIII.12]) Let f be a regular non-unt polynomal. The followng assertons are equvalent: () f s prmary () f = ug m for some unt u K, m 1 and G K[x] prme. () f = ug m + h for some u, g, h R[x], m 1 wth u unt, g basc rreducble and h M[x]. R[x] s not a unque factorsaton doman. However, polynomals n R[x] factor unquely nto prmary parwse coprme factors: Theorem 2.3 ([9, Theorem XIII.11]) Let f R[x] be a regular polynomal. Then f = uf 1 f 2 f s wth u R[x] a unt and f 1,..., f s R[x] regular prmary parwse coprme polynomals. The factors f are unque up to multplcaton by unts. The proof of the above theorem s constructve and uses Hensel lftng. We recall here the man steps. By Theorem 2.1(v) we may assume that f s monc. Frst we factor f n K[x], say f = F m Fs ms wth F K[x] rreducble and m 1 for = 1,..., s. Snce F m are coprme polynomals, one can use Hensel lftng to obtan a factorsaton f = f 1 f s wth f R[x], f = F m and f parwse coprme. By Theorem 2.2, f are prmary polynomals. Throughout the paper p wll be a prme number and Z p a the rng of ntegers modulo p a. The Galos feld wth p r elements s denoted GF(p r ). We denote by GR(p a, r) the Galos rng obtaned as Z p a[y]/ f wth f Z p a[y] a monc basc rreducble polynomal of degree r. Note that the characterstc of GR(p a, r) s p a. In ths paper we wll assume a 2, so that the Galos rng s not a feld. Note that Galos rngs are fnte local rngs. The maxmal deal of GR(p a, r) s M = p and the resdue feld s K = GF(p r ). We have c = c mod p for all c GR(p a, r). Every element of GR(p a, r) can be unquely wrtten as up wth 0 < a, unquely determned and u GR(p a, r) a unt, unque modulo p a. For any c GR(p a, r) f p c = 0 then c s dvsble by p a. All the prevous theorems hold n partcular for Galos rngs. Theorem 2.2 yelds n ths case: Corollary 2.4 Let f GR(p a, r)[x] be a monc polynomal. Then f s prmary ff f = g m + ph for some g, h R[x], m 1 wth g monc and basc rreducble. Note that the polynomals g and h n the corollary above are n general not unque. 3 Irreducblty crteron for prmary polynomals over Galos rngs We start wth a necessary (but not suffcent n general) condton for the reducblty of a prmary polynomal over a Galos rng. Theorem 3.1 Let f GR(p a, r)[x] be a monc prmary polynomal whch s not basc rreducble. Let g, h GR(p a, r)[x] and m 2 be such that f = g m + ph and g s monc basc rreducble. If f factors then h = 0 or g h. 3

6 Proof. Snce f factors, there are f 1, f 2 GR(p a, r)[x] monc non-constant polynomals such that f = f 1 f 2. Snce f = g m = f 1 f 2, we can wrte f = g m + ph for some m > 0, h GR(p a, r)[x] for = 1, 2 wth m 1 + m 2 = m. Wthout loss of generalty we can assume m 1 m 2. We have f = (g m 1 + ph 1 )(g m 2 + ph 2 ) = g m + pg m 1 (h 2 + h 1 g m 2 m 1 ) + p 2 h 1 h 2 = g m + ph Hence h = g m 1 (h2 + h 1 g m 2 m 1) and therefore we have ether h = 0 or g h as requred. The converse of the above theorem does not hold n general, as the followng example shows. However, f the Galos rng s of the form GR(p 2, r), the converse does hold, see Theorem 4.1. Example 3.2 Let f = (x+1) 4 +4x Z 8 [x]. Puttng g = x+1 and h = 2x we have f = g 4 +2h and g s monc basc rreducble. Note that h = 0. Moreover, any other polynomals g, h such that f = g 4 + 2h and g s monc basc rreducble are of the form g = x w for some w Z 8 and 2h = f (x w) 4 = 4x, and so h = 0. So f satsfes the concluson of Theorem 3.1. However, we wll show shortly that f s rreducble. So Theorem 3.1 gves a necessary, but not suffcent condton for a polynomal to factor. We show now that f s rreducble. It can be easly checked that f has no roots n Z 8, so t cannot have any monc factor of degree one. So we are left wth the possblty of f factorng nto two monc factors of degree two: f = ((x + 1) 2 + 2(Ax + B))((x + 1) 2 + 2(Cx + D)) for some A, B, C, D Z 8. By comparng lke coeffcents of these polynomals we obtan a system of equatons n the unknowns A, B, C and D whch has no solutons n Z 8. A suffcent condton for the rreducblty of a polynomal mmedately results from Theorem 3.1. It can be vewed as a generalsed Esensten crteron: Corollary 3.3 Let f GR(p a, r)[x] be a monc prmary polynomal whch s not basc rreducble. Let g, h GR(p a, r)[x] and m 2 be such that f = g m + ph and g s monc basc rreducble. If h 0 and g h then f s rreducble. Example 3.4 A polynomal of the form f = x s + p(a s 1 x s a 0 ) GR(p a, r) wth a 0 a unt s called an Esensten polynomal (see for example [9, p. 341]). Puttng g = x and h = a s 1 x s a 0, we see that h 0 and g h. So by Corollary 3.3, f s rreducble, as expected. If f s a polynomal such that f s square-free, the factorsaton of f nto prmary parwse coprme factors (gven by Theorem 2.3) s a factorsaton nto basc rreducble factors. If f s not square-free, some of the prmary factors may factor further. Below we gve a suffcent condton for all prmary factors n the factorsaton gven by Theorem 2.3 to be rreducble. Note that checkng ths condton does not requre factorng the polynomal. Proposton 3.5 Let f GR(p a, r)[x] be such that f s not square-free. Let f 1, f 2 be any polynomals n GR(p a, r)[x] such that f 1 s the square-free part of f and f = f 1 f 2. Let h GR(p a, r)[x] be such that ph = f f 1 f 2. If h 0 and h and f 2 are coprme then the factorsaton of f nto prmary parwse coprme factors (gven by Theorem 2.3) s a factorsaton nto rreducble factors. 4

7 Proof. Let f = s =1 Gm be the factorsaton of f nto rreducble polynomals n GF(p r ). Let g be any polynomals such that g = G. We have f 1 = s =1 G and f 2 = s =1 Gm 1, so f 1 = s =1 g +pw 1 and f 2 = s =1 gm 1 +pw 2 for some w 1, w 2 GR(p a, r). The factorsaton of f gven by Theorem 2.3 s of the form f = s =1 (gm + ph ) for some h GR(p a, r). To show that ths s a factorsaton nto rreducble factors t suffces (by Corollary 3.3) to show that for any for whch m > 1 we have h 0 and G h. By hypothess, h 0 and h and f 2 are coprme, so h s not dvsble by any of the G for whch m > 1. Computng f f 1 f 2 we obtan h = s =1 h j Gm j j w s 1 =1 Gm 1 w s 2 =1 G. Fx an such that m > 1. In the last equalty above, all the terms on the rght hand sde are dvsble by G except possbly for h j Gm j j. Snce the left hand sde s not dvsble by G we deduce h 0 and G h as requred. 4 Factorsaton of prmary polynomals over GR(p 2, r) From ths pont on, we wll restrct the coeffcent rng to a Galos rng of characterstc p 2. Theorem 3.1 can be mproved n ths settng, gvng a necessary and suffcent condton for a prmary polynomal to factor. Theorem 4.1 Let f GR(p 2, r)[x] be a monc prmary polynomal whch s not basc rreducble. Let g, h GR(p a, r)[x] and m 2 be such that f = g m + ph and g s monc basc rreducble. Then f factors f and only f h = 0 or g h. Proof. The drect mplcaton follows from Theorem 3.1. We prove the converse. If h = 0 then ph = 0 so f = g m and ths s a factorsaton of f nto rreducble factors. If h 0 let m 1 1 be maxmal such that g m 1 h and choose w so that h = g m 1 w. Snce p 2 = 0, we have ph = pg m 1 w. We thus obtan the factorsaton f = g m +ph = g m +pg m 1 w = g m 1 (g m m 1 +pw). By Corollary 3.3, g m m 1 + pw s rreducble snce w 0 and g w by constructon. So we factored f nto rreducble factors. The proof of the above theorem also yelds: Corollary 4.2 Let f GR(p 2, r)[x] be a monc prmary polynomal whch s not basc rreducble The followng assertons are equvalent: () f factors. () f has a basc rreducble factor. () for all g GR(p a, r)[x], f g s basc rreducble and g f then g f. When the Galos rng has characterstc p 2, the converse of Corollary 3.3 also holds: Corollary 4.3 Let f GR(p 2, r)[x] be a monc prmary polynomal whch s not basc rreducble. Let g, h GR(p a, r)[x] and m 2 be such that f = g m + ph and g s monc basc rreducble. Then f s rreducble f and only f h 0 and g h. If a polynomal n GR(p 2, r) factors, there are n general several possble factorsatons. We wll concentrate here on factorsatons that are maxmal n the sense that they contan the maxmum number of (not necessarly dstnct) factors. 5

8 Theorem 4.4 Let f GR(p 2, r)[x] be a monc prmary polynomal whch s not rreducble. Let m 2 and G GF(p r )[x] be the unquely determned elements such that f = G m n GF(p r )[x]. Then f admts a factorsaton nto monc rreducble factors of one (but not both) of the followng two types: () f = g m (1) for some g GR(p 2, r)[x] such that g s monc and g = G. () f = g m 1 (g m m 1 + pw) (2) for some g, w GR(p 2, r)[x] and 1 m 1 < m such that g s monc, g = G, g m m 1 + pw s rreducble and f p m then m m 1 2. The factorsatons gven above have the followng property: they are factorsatons of f nto the maxmum number of (not necessarly dstnct) rreducble factors, and among all possble factorsatons nto the maxmum number of rreducble factors, they consst of a mnmum number of dstnct factors. Moreover, all factorsatons of f nto monc rreducble factors havng ths property are factorsatons of type () or () and can be obtaned as follows: In case (), f p m then g s unquely determned; f p m then any monc g GR(p 2, r)[x] wth g = G satsfes (1). In case (), m 1 s unquely determned and for any monc g GR(p 2, r)[x] wth g = G there s a unque rreducble polynomal of the form g m m 1 +pw, wth w GR(p 2, r)[x], so that (2) s satsfed. Proof. The fact that f can be wrtten as n (1) or (2) follows from Theorem 4.1 and ts proof. We show that f f can be wrtten as n (2) but p m and m 1 = m 1, then f can be wrtten as n (1) for a dfferent choce of g. We have f = g m 1 (g + pw). Puttng g 2 = g + pu where u s any polynomal such that u = (m) 1 w one can verfy that f = g2 m. Assume now, for a contradcton, that f admts both a factorsaton of type (), say f = g1 m and a factorsaton of type (), say f = gm 1 (g m m 1 + pw). Snce g = g 1 = G, there s a u GR(p 2, r)[x] so that g 1 = g + pu. Hence g m + pg m 1 w = (g + pu) m = g m + pmg m 1 u, so w = mg m m1 1 u. We deduce that f p m then w = 0 and f p m then m m hence G w. But then, by Corollary 4.3, g m m 1 + pw would not be rreducble, so we obtan a contradcton. Next we prove the assertons about the number of factors. For () t s obvous that the number of (non-dstnct) factors s maxmal, and that the number of dstnct factors s one, therefore mnmal. For () consder an arbtrary factorsaton of f nto rreducble factors. It wll have the form f = s =1 (gk + pw ) wth 1 k 1 k 2... k s, s =1 k = m, w GR(p 2, r)[x] and g k + pw rreducble. From f = g m + p s =1 w g m k = g m + pg m 1 w we deduce g m k s s =1 w g k s k = g m 1 w. Hence m k s m 1. Snce s 1 =1 k = m k s m 1, we deduce that s m 1 +1, so m 1 +1 s the maxmal number of factors n any factorsaton of f. We also note that the equalty s = m 1 +1 (.e. factorsaton nto a maxmal number of factors) can only be reached when k 1 = k 2 =... = k s 1 = 1 and k s = m m 1. As factorsatons of the form () cannot be wrtten n the form (), the number of dstnct rreducble factors has to be at least two. Gven a factorsaton of f of type () or () we wll examne now what happens for a dfferent choce of g wth g = G. Let g 1 be another polynomal such that g 1 = G. There s a u GR(p 2, r)[x] so that g = g 1 + pu and pu 0. If f s n case () we have f = (g 1 + pu) m = g1 m+pmgm 1 1 u. Ths means that f p m then g 1 satsfes (1), otherwse t does not. If f s n case 6

9 () we have f = (g 1 +pu) m +p(g 1 +pu) m 1 w = g1 m +p(mgm 1 1 u+g m 1 1 w) = gm 1 1 (gm m 1 1 +pw 1 ), where we denoted w 1 = mg m 1 m 1 1 u + w. One can prove that g m m pw 1 s rreducble ether usng Corollary 4.3 or usng the fact that m s the maxmum number of factors of f, so any factorsaton nto m factors can only contan rreducble factors. It s easy to verfy that these constructons gve all the possble factorsatons satsfyng the stated requrements regardng the number of factors. We note that n the above theorem, f f s n case () or f f s n case () and p m, there are GF(p r ) deg(g) ways of choosng a monc g wth g = G. Hence, up to multplcaton by unts, there are GF(p r ) deg(g) factorsatons satsfyng the property n the theorem regardng the number of factors. Based on Theorems 4.1 and 4.4 we can now develop an algorthm for decdng f a prmary polynomal factors, and, n the affrmatve case, obtanng a factorsaton nto the maxmum number of rreducble factors. Algorthm 4.5 (Factorsaton of a prmary polynomal) Input: f GR(p 2, r)[x], a prmary polynomal. Output: A lst of pars ((f 1, m 1 ),..., (f s, m s )) so that f = f m fs ms and f are rreducble or one of the messages f s rreducble or f s basc rreducble. Note: The factorsaton has the maxmum number of factors; among all factorsatons nto the maxmum number of factors, ths has the mnmum number of dstnct factors. begn Determne G GF(p r )[x] and m 1 so that f = G m and G s rreducble. f m = 1 then return( f s basc rreducble ) Choose g GR(p 2, r)[x] monc so that g = G and determne h so that ph = f g m. f h = 0 then return(((g, m))) Determne the maxmum m 1 so that G m 1 h and determne w so that h = G m 1 w. f m 1 = 0 then return( f s rreducble ) f (p m) or (m 1 m 2) then return( ((g, m 1 ), (g m m 1 + pw, 1)) ) Choose u such that u = (m) 1 w. return(((g + pu, m))) end It s easy to see that the worst-case complexty of the algorthm above s quadratc n the degree of f. Once a factorsaton has been obtaned, one can easly wrte down all possble factorsatons havng the propertes n Theorem 4.4. Let us now apply the algorthm to an example: Example 4.6 Let f = x 3 + 6x Z 9 [x]. In Z 3 [x] we have f = x = (x + 1) 3. Hence f s prmary but t s not basc rreducble. Put g = x + 1 Z 9 [x], m = 3 and h = x 2 +2x+1. Snce h s dvsble by g 2 and p m, a factorsaton of f nto rreducble factors s f = (x + 1) 2 (x + 4). By takng all other possble values for g so that g = x + 1 we get all the other factorsatons of f of ths type, namely f = (x + 4) 2 (x + 7) and f = (x + 7) 2 (x + 1). Note that when vewed as a polynomal over the 3-adc numbers, f s rreducble (for example f has no roots n Z 27 so t s rreducble n Z 27 already). Hence none of these factorsatons could be obtaned by projectng to Z 9 [x] the factorsaton of f over the 3-adc numbers. 7

10 Usng Theorem 4.4 and ts proof, one can also obtan all the factorsatons of a prmary polynomal nto the maxmum number of rreducble factors (wthout the restrcton on havng a mnmal number of dstnct factors): Corollary 4.7 Let f GR(p 2, r)[x] be a monc prmary polynomal whch s not rreducble. () Assume f admts a factorsaton f = g m as n Theorem 4.4(). Then f = m =1 (g + pw ) wth w GR(p 2, r)[x] arbtrary of degree less than deg(g), for = 1,..., m 1 and w m = m 1 =1 w, gves all the possble factorsatons of f nto a maxmum number of monc rreducble factors. () If f admts a factorsaton f = g m 1 (g m m 1 + pw) as n Theorem 4.4(), then f = ( m 1 =1 (g + pw ))(g m m 1 + pw m1 +1) wth w GR(p 2, r)[x] arbtrary of degree less than deg(g) for = 1,..., m 1, and w m1 +1 = w g m m 1 1 m 1 =1 w, gves all the factorsatons of f nto a maxmum number of monc rreducble factors. Proof. One can mmedately verfy that the formulae above are ndeed factorsatons of f nto the maxmum number of factors, hence all factors wll be rreducble. Next we have to show that we obtan ndeed all the possble factorsatons nto a maxmum number of factors. For (), ths s mmedate. For (), we noted n the proof of Theorem 4.4 that (wth the notatons from that proof), any factorsaton nto a maxmum number of factors has to satsfy k 1 = k 2 =... = k s 1 = 1 and k s = m m 1. Remark 4.8 Polynomals n GR(p 2, r)[x] may also factor nto fewer than the maxmum number of rreducble factors gven by Theorem 4.4. For example, f f = g m wth m 4, we can wrte f = (g k + pu)(g k pu)g m 2k for any 2 k m/2 and any u GR(p 2, r)[x] so that deg(u) < deg(g k ), u 0 and g u. Ths s a factorsaton nto m 2k + 2 < m rreducble factors. For example we have the two factorsatons x 4 = (x 2 + 2) 2 n Z 4 [x] and x s rreducble. We wll not examne ths type of factorsatons any further n ths paper. Usng Corollary 4.3, one can easly show that the converse of Proposton 3.5 holds for Galos rngs of characterstc p 2 : Corollary 4.9 Let f GR(p 2, r)[x] be such that f s not square-free. Let f 1, f 2 be any polynomals n GR(p 2, r)[x] such that f 1 s the square-free part of f and f = f 1 f 2. Let h GR(p 2, r)[x] be such that ph = f f 1 f 2. The factorsaton of f nto prmary parwse coprme factors (gven by Theorem 2.3) s a factorsaton nto rreducble factors f and only f h 0 and h and f 2 are coprme. We note an nterestng connecton between the factorsaton of a polynomal f and GR(p a, r)[x]/ f beng a prncpal deal rng. Theorem 4.10 Let f GR(p a, r)[x]. () If GR(p a, r)[x]/ f s a prncpal deal rng then the factorsaton of f nto prmary parwse coprme factors (gven by Theorem 2.3) s a factorsaton nto rreducble factors. () When a = 2, GR(p 2, r)[x]/ f s a prncpal deal rng f and only f the factorsaton of f nto prmary parwse coprme factors (gven by Theorem 2.3) s a factorsaton nto rreducble factors. 8

11 Proof. Wth the notatons of Proposton 3.5, we have that GR(p a, r)[x]/ f s a prncpal deal rng f and only f h 0 and h and f 2 are coprme (see [3, Theorem 4]; also [11, Theorem 3.2],[12]). The result now follows from Proposton 3.5 for () and from Corollary 4.9 for (). Remark 4.11 Note that the converse of pont () n the theorem above does not hold for a > 2. For example, one can check that although f = (x + 1) 4 + 4x Z 8 [x] s prmary and rreducble (see Example 3.2), Z 8 [x]/ f s not a prncpal deal rng (for example the deal x + 1, 2 s not prncpal). 5 Applcaton: factorng x n 1 and x n + 1 In ths secton we determne factorsatons of x n 1 and of x n + 1 nto a maxmal number of rreducble factors over GR(p 2, r)[x]. The polynomal x n 1 s mportant for numerous applcatons. Our motvaton comes from codng theory, where cyclc codes over a Galos rng are deals n GR(p a, r)[x]/ x n 1. Negacyclc codes are deals n GR(p a, r)[x]/ x n + 1. One usually assumes that n s not dvsble by p, but the case when p n, yeldng the so-called repeated-roots codes, s also of nterest. When n s not dvsble by p, the polynomal x n 1 has no multple factors over GF(p r ). Hensel lftng wll produce then a unque factorsaton of x n 1 over GR(p a, r)[x] wth all factors basc rreducble. The same happens for x n + 1. Factorng x n 1 (or x n + 1) s more complcated when p n. Here we deal wth ths case n rngs of the form GR(p 2, r) (these rngs nclude n partcular Z 4, whch s an mportant rng for codng theory applcatons). Theorem 5.1 Let x n 1 GR(p 2, r)[x] and assume p n. Wrte n as n = kp b wth b 1 and p k. Let h GR(p 2, r)[x] be any polynomal such that { 1 f p = 2 h = p 2 =1 ( j=1 j 1 )x kpb 1 f p > 2 Then () x n 1 = (x k 1) pb 1 ((x k 1) (p 1)pb 1 + ph) and h s relatvely prme to x k 1 n GF(p r )[x]. () Let x k 1 = s =1 f be the factorsaton of x k 1 nto basc rreducble factors over GR(p 2, r)[x] and let w GR(p 2, r)[x] be such that (x k 1) (p 1)pb 1 + ph = s (p 1)pb 1 =1 (f + pw ) s the factorsaton of (x k 1) (p 1)pb 1 +ph nto prmary parwse coprme factors. Then x n 1 = s =1 f pb 1 (f (p 1)pb 1 + pw ) (3) s a factorsaton of x n 1 nto the maxmum number of (not necessarly dstnct) rreducble factors; among all possble factorsatons nto the maxmum number of rreducble factors, the factorsaton above conssts of the mnmum number of dstnct factors. 9

12 Proof. () In GF(p r )[x] we have x n 1 = (x k 1) pb. Hence n GR(p 2, r) we have x n 1 = (x k 1) pb + pt for some polynomal t whch we wll now determne. ( ) p For any 0 < j < p b b, we know by Kummer s theorem that s dvsble by p j b c (and ( by ) no hgher power of p) where c s the hghest exponent so that p c j. So n partcular p b 0 mod p j 2 for all values 0 < j < p b for whch j s not dvsble by p b 1. When j s ( ) p of the form j = p b 1 b wth 0 < < p, s dvsble by p but not by p 2. We wll treat the case p = 2 frst: p b 1 2t = x n 1 (x k 1) 2b = x n 1 (x n + 2x k2b 1 + 1) = 2(x k2b 1 + 1) = 2(x k 1) 2b 1. Therefore x n 1 can be wrtten as n the theorem, wth h = 1 n ths case. Now we assume p > 2. We have pt = x n 1 (x k 1) pb = x n 1 p ( p b =0 ( p b By Lemma 6.1 n the Appendx, =1 p b 1 p b 1 ) p 1 ( p b x kpb 1 ( 1) (p )pb 1 = =1 p b 1 ) pc mod p 2 where c = ( 1) 1 1. Hence p 1 p 1 t = ( 1) 1 1 x kpb 1 ( 1) p = 1 x kpb 1. =1 ) x kpb 1 ( 1) p. In GF(p r )[x] we dvde t by (x k 1) pb 1 = x kpb 1 1. We obtan the remander p 1 =1 1 = p 1 =1 = p(p 1)/2 0 mod p (as 1 wll take all values between 1 and p 1 when vares from 1 to p 1) and the quotent p 2 h = p 1 =0 j=+1 j 1 x kpb 1 = p 2 =1 j=1 j 1 x kpb 1 (here agan we used the fact that p 1 =1 1 0 mod p). It remans to show that h s coprme to x k 1. Assume they had a common factor. Then they would have a common root ξ n a sutable extenson feld. As ξ s a root of x k 1, we have ξ k = 1. Evaluatng h at ξ we obtan p 2 h(ξ) = p 1 =0 j=+1 p 1 j 1 = j=1 p 1 jj 1 = j=1 Hence we obtan a contradcton, as ξ cannot be a root of h. () By Corollary 4.9, (x k 1) (p 1)pb 1 + ph = s =1 1 = (p 1) = 1 (f (p 1)pb 1 + pw ) s the factorsaton of (x k 1) (p 1)pb 1 + ph nto rreducble factors, as h s coprme to x k 1. Hence (3) s a factorsaton nto rreducble factors. It remans to prove the assertons about the number of rreducble factors. The factorsaton of x n 1 nto monc prmary parwse coprme factors s unque (Theorem 2.3) and 10

13 from (3) there are s prmary parwse coprme factors, namely f pb 1 + pw ), for = 1,..., s. By Theorem 4.4, each of these factors s factored n (3) nto a maxmal number of rreducble factors, and the number of dstnct factors s mnmal among all such factorsatons. (f (p 1)pb 1 Usng smlar technques one can determne a factorsaton of x n + 1. Note that the cases p = 2 and p > 2 dffer more substantally here. Theorem 5.2 Let x n + 1 GR(p 2, r)[x] and assume p n. () If p = 2 then the factorsaton of x n + 1 nto prmary parwse coprme factors n GR(2 2, r)[x] (gven by Theorem 2.3) s also a factorsaton nto rreducble factors. () Let p > 2. Wrte n as n = kp b wth b 1 and p k. Let h be any polynomal such that h = p 2 =1 ( 1) ( j=1 j 1 )x kpb 1. Then x n + 1 = (x k + 1) pb 1 ((x k + 1) (p 1)pb 1 + ph) and h s relatvely prme to x k + 1 n GF(p r )[x]. Let x k + 1 = s =1 f be the factorsaton of x k + 1 nto basc rreducble factors over GR(p 2, r)[x] and let w GR(p 2, r)[x] be such that (x k + 1) (p 1)pb 1 + ph = s (p 1)pb 1 (f + pw ) s the factorsaton of (x k + 1) (p 1)pb 1 + ph =1 nto prmary parwse coprme factors. Then x n + 1 = s =1 f pb 1 (f (p 1)pb 1 + pw ) (4) s a factorsaton of x n + 1 nto the maxmum number of (not necessarly dstnct) rreducble factors; among all possble factorsatons nto the maxmum number of rreducble factors, the factorsaton above conssts of the mnmum number of dstnct factors. Proof. We wll use the same notatons as n the proof of Theorem 5.1. () Assume p = 2. Then x n + 1 = (x k + 1) 2b + 2t and 2t = 2x k2b 1. Obvously t = x k2b 1 s non-zero and coprme to x k + 1. Hence by Corollary 4.9, the factorsaton of x n + 1 nto prmary coprme factors s also a factorsaton nto rreducble factors. () Assume p > 2. We have x n + 1 = (x k + 1) pb + pt wth p 1 t = ( 1) 1 1 x kpb 1 = =1 p 1 ( 1) 1 x kpb 1. When dvdng t by (x k + 1) pb 1 = x kpb n GF(p r ) we obtan the remander zero and the quotent h, whch one can check that s relatvely prme to x k + 1. The rest of the proof s smlar to the proof of Theorem 5.1. =1 Remark 5.3 We note that the results of Theorems 5.1 and 5.2 together wth Corollary 4.9 mply n partcular that GR(p 2, r)[x]/ x n 1 s not a prncpal deal rng whereas GR(p 2, r)[x]/ x n + 1 s a prncpal deal rng f p = 2 but t s not a prncpal deal rng when p > 2. We retreve thus partcular cases of [11, Theorem 3.4], [12]. Acknowledgement I would lke to thank Serpl Acar for her encouragement whle wrtng ths paper. 11

14 6 Appendx Lemma ( 6.1 Let ) p be ( a prme ) number, b 1 and 0 < < p. We have: p b p () p b 1 (mod p b ) ( ) p () Let c = /p Z (the dvson s exact). Then c mod p = ( 1) 1 1 n Z p. Proof. () We wll use the usual formula ( n k that are dvsble by p b 1 : ( ) p b p b 1 = pb (p 1)p b 1... (p + 1)p b 1 p b 1 2p b 1... p b 1 ) = n (n 1)... (n k+1) k!, separatng the factors (pb 1)(p b 2)... (p b p b 1 + 1)(p b p b 1 1)... (p b p b 1 + 1) (p b 1 1)(p b 1 + 1)... (p b 1 1) We denote by A and B the frst and the second fracton above, respectvely. For A we have n Z ( ) p(p 1)... (p + 1) p A = =.! Obvously A s dvsble by p. So for evaluatng AB mod p b t suffces to evaluate B mod p b 1. One can check that, modulo p b 1, both the numerator and the denomnator of B equal (p b 1 1)!, so B mod p b 1 = 1. () We have c = (p 1)(p 2)... (p +1)!, so c mod p = ( 1)( 2)... ( ( 1))! = ( 1) 1 1 n Z p. References [1] E.R. Berlekamp. Factorng polynomals over large fnte felds. Math. Comp., 24: , [2] D.G. Cantor and Gordon D.M. Factorng polynomals over p-adc felds. In W. Bosma, edtor, Algorthmc Number Theory, 4th Internatonal Symposum, ANTS-IV, Leden, The Netherlands, July 2-7, 2000, Proceedngs, volume 1838 of Lecture Notes n Computer Scence. Sprnger, [3] J. Cazaran and A.V. Kelarev. Generators and weghts of polynomal codes. Archv der Mathematk, 69: , [4] A.L. Chstov. Effcent factorsaton of polynomals over local felds. Sovet. math. Dokl., 35: , [5] A.L. Chstov. Algorthm of polynomal complexty for factorng polynomals over local felds. J. Math. Scences, 70: , [6] D. Ford, S. Paul, and X-F. Roblot. A fast algorthm for polynomal factorzaton over Q p. J. Th. Nombres Bordeaux, 14: ,

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