Factoring polynomials over Z4 and over certain Galois rings
|
|
- Adelia Adams
- 5 years ago
- Views:
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: ,
15 [7] D.J. Ford. The constructon of maxmal orders over a Dedeknd doman. J. Symb. Comput., 4:69 75, [8] A. R. Hammons, Jr., P. V. Kumar, A. R. Calderbank, N. J. A. Sloane, and P. Solé. The Z 4 lnearty of Kerdock, Preparata, Goethals and related codes. IEEE Trans. Inform. Theory, 40: , [9] B. R. McDonald. Fnte Rngs wth Identty. Marcel Dekker, New York, [10] S. Paul. Factorng polynomals over local felds. J. Symb. Comput., 32: , [11] A. Sălăgean. Repeated-root cyclc and negacyclc codes over a fnte chan rng. In Proceedngs of the Workshop on Codng and Cryptography, Pars, March, pages , [12] A. Sălăgean. Repeated-root cyclc and negacyclc codes over a fnte chan rng. Dscrete Appled Mathematcs, to appear. [13] J. von zur Gathen and S. Hartleb. Factorng modular polynomals. J. Symbolc Computaton, 26: ,
FINITELY-GENERATED MODULES OVER A PRINCIPAL IDEAL DOMAIN
FINITELY-GENERTED MODULES OVER PRINCIPL IDEL DOMIN EMMNUEL KOWLSKI Throughout ths note, s a prncpal deal doman. We recall the classfcaton theorem: Theorem 1. Let M be a fntely-generated -module. (1) There
More informationFACTORIZATION IN KRULL MONOIDS WITH INFINITE CLASS GROUP
C O L L O Q U I U M M A T H E M A T I C U M VOL. 80 1999 NO. 1 FACTORIZATION IN KRULL MONOIDS WITH INFINITE CLASS GROUP BY FLORIAN K A I N R A T H (GRAZ) Abstract. Let H be a Krull monod wth nfnte class
More informationDISCRIMINANTS AND RAMIFIED PRIMES. 1. Introduction A prime number p is said to be ramified in a number field K if the prime ideal factorization
DISCRIMINANTS AND RAMIFIED PRIMES KEITH CONRAD 1. Introducton A prme number p s sad to be ramfed n a number feld K f the prme deal factorzaton (1.1) (p) = po K = p e 1 1 peg g has some e greater than 1.
More informationTHE CHINESE REMAINDER THEOREM. We should thank the Chinese for their wonderful remainder theorem. Glenn Stevens
THE CHINESE REMAINDER THEOREM KEITH CONRAD We should thank the Chnese for ther wonderful remander theorem. Glenn Stevens 1. Introducton The Chnese remander theorem says we can unquely solve any par of
More informationThe Ramanujan-Nagell Theorem: Understanding the Proof By Spencer De Chenne
The Ramanujan-Nagell Theorem: Understandng the Proof By Spencer De Chenne 1 Introducton The Ramanujan-Nagell Theorem, frst proposed as a conjecture by Srnvasa Ramanujan n 1943 and later proven by Trygve
More informationALGEBRA HW 7 CLAY SHONKWILER
ALGEBRA HW 7 CLAY SHONKWILER 1 Whch of the followng rngs R are dscrete valuaton rngs? For those that are, fnd the fracton feld K = frac R, the resdue feld k = R/m (where m) s the maxmal deal), and a unformzer
More informationMath 261 Exercise sheet 2
Math 261 Exercse sheet 2 http://staff.aub.edu.lb/~nm116/teachng/2017/math261/ndex.html Verson: September 25, 2017 Answers are due for Monday 25 September, 11AM. The use of calculators s allowed. Exercse
More informationREDUCTION MODULO p. We will prove the reduction modulo p theorem in the general form as given by exercise 4.12, p. 143, of [1].
REDUCTION MODULO p. IAN KIMING We wll prove the reducton modulo p theorem n the general form as gven by exercse 4.12, p. 143, of [1]. We consder an ellptc curve E defned over Q and gven by a Weerstraß
More informationDifference Equations
Dfference Equatons c Jan Vrbk 1 Bascs Suppose a sequence of numbers, say a 0,a 1,a,a 3,... s defned by a certan general relatonshp between, say, three consecutve values of the sequence, e.g. a + +3a +1
More informationa b a In case b 0, a being divisible by b is the same as to say that
Secton 6.2 Dvsblty among the ntegers An nteger a ε s dvsble by b ε f there s an nteger c ε such that a = bc. Note that s dvsble by any nteger b, snce = b. On the other hand, a s dvsble by only f a = :
More informationThe Order Relation and Trace Inequalities for. Hermitian Operators
Internatonal Mathematcal Forum, Vol 3, 08, no, 507-57 HIKARI Ltd, wwwm-hkarcom https://doorg/0988/mf088055 The Order Relaton and Trace Inequaltes for Hermtan Operators Y Huang School of Informaton Scence
More informationFoundations of Arithmetic
Foundatons of Arthmetc Notaton We shall denote the sum and product of numbers n the usual notaton as a 2 + a 2 + a 3 + + a = a, a 1 a 2 a 3 a = a The notaton a b means a dvdes b,.e. ac = b where c s an
More informationFinding Primitive Roots Pseudo-Deterministically
Electronc Colloquum on Computatonal Complexty, Report No 207 (205) Fndng Prmtve Roots Pseudo-Determnstcally Ofer Grossman December 22, 205 Abstract Pseudo-determnstc algorthms are randomzed search algorthms
More informationTHERE ARE NO POINTS OF ORDER 11 ON ELLIPTIC CURVES OVER Q.
THERE ARE NO POINTS OF ORDER 11 ON ELLIPTIC CURVES OVER Q. IAN KIMING We shall prove the followng result from [2]: Theorem 1. (Bllng-Mahler, 1940, cf. [2]) An ellptc curve defned over Q does not have a
More informationinv lve a journal of mathematics 2008 Vol. 1, No. 1 Divisibility of class numbers of imaginary quadratic function fields
nv lve a journal of mathematcs Dvsblty of class numbers of magnary quadratc functon felds Adam Merberg mathematcal scences publshers 2008 Vol. 1, No. 1 INVOLVE 1:1(2008) Dvsblty of class numbers of magnary
More informationDIFFERENTIAL FORMS BRIAN OSSERMAN
DIFFERENTIAL FORMS BRIAN OSSERMAN Dfferentals are an mportant topc n algebrac geometry, allowng the use of some classcal geometrc arguments n the context of varetes over any feld. We wll use them to defne
More information5 The Rational Canonical Form
5 The Ratonal Canoncal Form Here p s a monc rreducble factor of the mnmum polynomal m T and s not necessarly of degree one Let F p denote the feld constructed earler n the course, consstng of all matrces
More informationLecture 5 Decoding Binary BCH Codes
Lecture 5 Decodng Bnary BCH Codes In ths class, we wll ntroduce dfferent methods for decodng BCH codes 51 Decodng the [15, 7, 5] 2 -BCH Code Consder the [15, 7, 5] 2 -code C we ntroduced n the last lecture
More information12 MATH 101A: ALGEBRA I, PART C: MULTILINEAR ALGEBRA. 4. Tensor product
12 MATH 101A: ALGEBRA I, PART C: MULTILINEAR ALGEBRA Here s an outlne of what I dd: (1) categorcal defnton (2) constructon (3) lst of basc propertes (4) dstrbutve property (5) rght exactness (6) localzaton
More informationLECTURE V. 1. More on the Chinese Remainder Theorem We begin by recalling this theorem, proven in the preceeding lecture.
LECTURE V EDWIN SPARK 1. More on the Chnese Remander Theorem We begn by recallng ths theorem, proven n the preceedng lecture. Theorem 1.1 (Chnese Remander Theorem). Let R be a rng wth deals I 1, I 2,...,
More informationAffine transformations and convexity
Affne transformatons and convexty The purpose of ths document s to prove some basc propertes of affne transformatons nvolvng convex sets. Here are a few onlne references for background nformaton: http://math.ucr.edu/
More informationAPPENDIX A Some Linear Algebra
APPENDIX A Some Lnear Algebra The collecton of m, n matrces A.1 Matrces a 1,1,..., a 1,n A = a m,1,..., a m,n wth real elements a,j s denoted by R m,n. If n = 1 then A s called a column vector. Smlarly,
More informationPolynomials. 1 What is a polynomial? John Stalker
Polynomals John Stalker What s a polynomal? If you thnk you already know what a polynomal s then skp ths secton. Just be aware that I consstently wrte thngs lke p = c z j =0 nstead of p(z) = c z. =0 You
More informationwhere a is any ideal of R. Lemma Let R be a ring. Then X = Spec R is a topological space. Moreover the open sets
11. Schemes To defne schemes, just as wth algebrac varetes, the dea s to frst defne what an affne scheme s, and then realse an arbtrary scheme, as somethng whch s locally an affne scheme. The defnton of
More informationRestricted divisor sums
ACTA ARITHMETICA 02 2002) Restrcted dvsor sums by Kevn A Broughan Hamlton) Introducton There s a body of work n the lterature on varous restrcted sums of the number of dvsors of an nteger functon ncludng
More informationwhere a is any ideal of R. Lemma 5.4. Let R be a ring. Then X = Spec R is a topological space Moreover the open sets
5. Schemes To defne schemes, just as wth algebrac varetes, the dea s to frst defne what an affne scheme s, and then realse an arbtrary scheme, as somethng whch s locally an affne scheme. The defnton of
More informationSL n (F ) Equals its Own Derived Group
Internatonal Journal of Algebra, Vol. 2, 2008, no. 12, 585-594 SL n (F ) Equals ts Own Derved Group Jorge Macel BMCC-The Cty Unversty of New York, CUNY 199 Chambers street, New York, NY 10007, USA macel@cms.nyu.edu
More informationSmarandache-Zero Divisors in Group Rings
Smarandache-Zero Dvsors n Group Rngs W.B. Vasantha and Moon K. Chetry Department of Mathematcs I.I.T Madras, Chenna The study of zero-dvsors n group rngs had become nterestng problem snce 1940 wth the
More informationThe internal structure of natural numbers and one method for the definition of large prime numbers
The nternal structure of natural numbers and one method for the defnton of large prme numbers Emmanul Manousos APM Insttute for the Advancement of Physcs and Mathematcs 3 Poulou str. 53 Athens Greece Abstract
More informationPerron Vectors of an Irreducible Nonnegative Interval Matrix
Perron Vectors of an Irreducble Nonnegatve Interval Matrx Jr Rohn August 4 2005 Abstract As s well known an rreducble nonnegatve matrx possesses a unquely determned Perron vector. As the man result of
More informationAlgebraic properties of polynomial iterates
Algebrac propertes of polynomal terates Alna Ostafe Department of Computng Macquare Unversty 1 Motvaton 1. Better and cryptographcally stronger pseudorandom number generators (PRNG) as lnear constructons
More information3.1 Expectation of Functions of Several Random Variables. )' be a k-dimensional discrete or continuous random vector, with joint PMF p (, E X E X1 E X
Statstcs 1: Probablty Theory II 37 3 EPECTATION OF SEVERAL RANDOM VARIABLES As n Probablty Theory I, the nterest n most stuatons les not on the actual dstrbuton of a random vector, but rather on a number
More informationCollege of Computer & Information Science Fall 2009 Northeastern University 20 October 2009
College of Computer & Informaton Scence Fall 2009 Northeastern Unversty 20 October 2009 CS7880: Algorthmc Power Tools Scrbe: Jan Wen and Laura Poplawsk Lecture Outlne: Prmal-dual schema Network Desgn:
More informationExample: (13320, 22140) =? Solution #1: The divisors of are 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 27, 30, 36, 41,
The greatest common dvsor of two ntegers a and b (not both zero) s the largest nteger whch s a common factor of both a and b. We denote ths number by gcd(a, b), or smply (a, b) when there s no confuson
More informationn α j x j = 0 j=1 has a nontrivial solution. Here A is the n k matrix whose jth column is the vector for all t j=0
MODULE 2 Topcs: Lnear ndependence, bass and dmenson We have seen that f n a set of vectors one vector s a lnear combnaton of the remanng vectors n the set then the span of the set s unchanged f that vector
More informationarxiv: v6 [math.nt] 23 Aug 2016
A NOTE ON ODD PERFECT NUMBERS JOSE ARNALDO B. DRIS AND FLORIAN LUCA arxv:03.437v6 [math.nt] 23 Aug 206 Abstract. In ths note, we show that f N s an odd perfect number and q α s some prme power exactly
More informationMore metrics on cartesian products
More metrcs on cartesan products If (X, d ) are metrc spaces for 1 n, then n Secton II4 of the lecture notes we defned three metrcs on X whose underlyng topologes are the product topology The purpose of
More informationSection 3.6 Complex Zeros
04 Chapter Secton 6 Comple Zeros When fndng the zeros of polynomals, at some pont you're faced wth the problem Whle there are clearly no real numbers that are solutons to ths equaton, leavng thngs there
More informationRemarks on the Properties of a Quasi-Fibonacci-like Polynomial Sequence
Remarks on the Propertes of a Quas-Fbonacc-lke Polynomal Sequence Brce Merwne LIU Brooklyn Ilan Wenschelbaum Wesleyan Unversty Abstract Consder the Quas-Fbonacc-lke Polynomal Sequence gven by F 0 = 1,
More informationFixed points of IA-endomorphisms of a free metabelian Lie algebra
Proc. Indan Acad. Sc. (Math. Sc.) Vol. 121, No. 4, November 2011, pp. 405 416. c Indan Academy of Scences Fxed ponts of IA-endomorphsms of a free metabelan Le algebra NAIME EKICI 1 and DEMET PARLAK SÖNMEZ
More informationNOTES ON SIMPLIFICATION OF MATRICES
NOTES ON SIMPLIFICATION OF MATRICES JONATHAN LUK These notes dscuss how to smplfy an (n n) matrx In partcular, we expand on some of the materal from the textbook (wth some repetton) Part of the exposton
More informationFACTORING POLYNOMIALS OVER FINITE FIELDS USING BALANCE TEST CHANDAN SAHA
FACTORING POLYNOMIALS OVER FINITE FIELDS USING BALANCE TEST CHANDAN SAHA Department of Computer Scence and Engneerng Indan Insttute of Technology Kanpur E-mal address: csaha@cse.tk.ac.n Abstract. We study
More informationk(k 1)(k 2)(p 2) 6(p d.
BLOCK-TRANSITIVE 3-DESIGNS WITH AFFINE AUTOMORPHISM GROUP Greg Gamble Let X = (Z p d where p s an odd prme and d N, and let B X, B = k. Then t was shown by Praeger that the set B = {B g g AGL d (p} s the
More informationShort running title: A generating function approach A GENERATING FUNCTION APPROACH TO COUNTING THEOREMS FOR SQUARE-FREE POLYNOMIALS AND MAXIMAL TORI
Short runnng ttle: A generatng functon approach A GENERATING FUNCTION APPROACH TO COUNTING THEOREMS FOR SQUARE-FREE POLYNOMIALS AND MAXIMAL TORI JASON FULMAN Abstract. A recent paper of Church, Ellenberg,
More informationChristian Aebi Collège Calvin, Geneva, Switzerland
#A7 INTEGERS 12 (2012) A PROPERTY OF TWIN PRIMES Chrstan Aeb Collège Calvn, Geneva, Swtzerland chrstan.aeb@edu.ge.ch Grant Carns Department of Mathematcs, La Trobe Unversty, Melbourne, Australa G.Carns@latrobe.edu.au
More informationLectures - Week 4 Matrix norms, Conditioning, Vector Spaces, Linear Independence, Spanning sets and Basis, Null space and Range of a Matrix
Lectures - Week 4 Matrx norms, Condtonng, Vector Spaces, Lnear Independence, Spannng sets and Bass, Null space and Range of a Matrx Matrx Norms Now we turn to assocatng a number to each matrx. We could
More informationIN POLYNOMIAL RINGS IN ONE VARIABLE OVER DEDEKIND DOMAINS W.W. ADAMS AND P. LOUSTAUNAU
GR OBNER BASES AND PRIMARY DECOMPOSITION IN POLYNOMIAL RINGS IN ONE VARIABLE OVER DEDEKIND DOMAINS W.W. ADAMS AND P. LOUSTAUNAU Abstract. Let D be a Dedeknd doman wth quotent eld K, let x be a sngle varable,
More informationGraph Reconstruction by Permutations
Graph Reconstructon by Permutatons Perre Ille and Wllam Kocay* Insttut de Mathémathques de Lumny CNRS UMR 6206 163 avenue de Lumny, Case 907 13288 Marselle Cedex 9, France e-mal: lle@ml.unv-mrs.fr Computer
More informationProblem Solving in Math (Math 43900) Fall 2013
Problem Solvng n Math (Math 43900) Fall 2013 Week four (September 17) solutons Instructor: Davd Galvn 1. Let a and b be two nteger for whch a b s dvsble by 3. Prove that a 3 b 3 s dvsble by 9. Soluton:
More informationLinear Algebra and its Applications
Lnear Algebra and ts Applcatons 4 (00) 5 56 Contents lsts avalable at ScenceDrect Lnear Algebra and ts Applcatons journal homepage: wwwelsevercom/locate/laa Notes on Hlbert and Cauchy matrces Mroslav Fedler
More informationDifferential Polynomials
JASS 07 - Polynomals: Ther Power and How to Use Them Dfferental Polynomals Stephan Rtscher March 18, 2007 Abstract Ths artcle gves an bref ntroducton nto dfferental polynomals, deals and manfolds and ther
More informationISSN: ISO 9001:2008 Certified International Journal of Engineering and Innovative Technology (IJEIT) Volume 3, Issue 1, July 2013
ISSN: 2277-375 Constructon of Trend Free Run Orders for Orthogonal rrays Usng Codes bstract: Sometmes when the expermental runs are carred out n a tme order sequence, the response can depend on the run
More informationFACTORING POLYNOMIALS OVER FINITE FIELDS USING BALANCE TEST CHANDAN SAHA
FACTORING POLYNOMIALS OVER FINITE FIELDS USING BALANCE TEST CHANDAN SAHA Department of Computer Scence and Engneerng Indan Insttute of Technology Kanpur E-mal address: csaha@cse.tk.ac.n Abstract. We study
More informationProblem Set 9 Solutions
Desgn and Analyss of Algorthms May 4, 2015 Massachusetts Insttute of Technology 6.046J/18.410J Profs. Erk Demane, Srn Devadas, and Nancy Lynch Problem Set 9 Solutons Problem Set 9 Solutons Ths problem
More informationTHE CLASS NUMBER THEOREM
THE CLASS NUMBER THEOREM TIMUR AKMAN-DUFFY Abstract. In basc number theory we encounter the class group (also known as the deal class group). Ths group measures the extent that a rng fals to be a prncpal
More informationarxiv: v1 [math.co] 12 Sep 2014
arxv:1409.3707v1 [math.co] 12 Sep 2014 On the bnomal sums of Horadam sequence Nazmye Ylmaz and Necat Taskara Department of Mathematcs, Scence Faculty, Selcuk Unversty, 42075, Campus, Konya, Turkey March
More informationAn Introduction to Morita Theory
An Introducton to Morta Theory Matt Booth October 2015 Nov. 2017: made a few revsons. Thanks to Nng Shan for catchng a typo. My man reference for these notes was Chapter II of Bass s book Algebrac K-Theory
More informationDeterminants Containing Powers of Generalized Fibonacci Numbers
1 2 3 47 6 23 11 Journal of Integer Sequences, Vol 19 (2016), Artcle 1671 Determnants Contanng Powers of Generalzed Fbonacc Numbers Aram Tangboonduangjt and Thotsaporn Thanatpanonda Mahdol Unversty Internatonal
More informationA summation on Bernoulli numbers
Journal of Number Theory 111 (005 37 391 www.elsever.com/locate/jnt A summaton on Bernoull numbers Kwang-Wu Chen Department of Mathematcs and Computer Scence Educaton, Tape Muncpal Teachers College, No.
More informationGames of Threats. Elon Kohlberg Abraham Neyman. Working Paper
Games of Threats Elon Kohlberg Abraham Neyman Workng Paper 18-023 Games of Threats Elon Kohlberg Harvard Busness School Abraham Neyman The Hebrew Unversty of Jerusalem Workng Paper 18-023 Copyrght 2017
More informationAffine and Riemannian Connections
Affne and Remannan Connectons Semnar Remannan Geometry Summer Term 2015 Prof Dr Anna Wenhard and Dr Gye-Seon Lee Jakob Ullmann Notaton: X(M) space of smooth vector felds on M D(M) space of smooth functons
More informationMaximizing the number of nonnegative subsets
Maxmzng the number of nonnegatve subsets Noga Alon Hao Huang December 1, 213 Abstract Gven a set of n real numbers, f the sum of elements of every subset of sze larger than k s negatve, what s the maxmum
More informationOn the irreducibility of a truncated binomial expansion
On the rreducblty of a truncated bnomal expanson by Mchael Flaseta, Angel Kumchev and Dmtr V. Pasechnk 1 Introducton For postve ntegers k and n wth k n 1, defne P n,k (x = =0 ( n x. In the case that k
More information2.3 Nilpotent endomorphisms
s a block dagonal matrx, wth A Mat dm U (C) In fact, we can assume that B = B 1 B k, wth B an ordered bass of U, and that A = [f U ] B, where f U : U U s the restrcton of f to U 40 23 Nlpotent endomorphsms
More informationCaps and Colouring Steiner Triple Systems
Desgns, Codes and Cryptography, 13, 51 55 (1998) c 1998 Kluwer Academc Publshers, Boston. Manufactured n The Netherlands. Caps and Colourng Stener Trple Systems AIDEN BRUEN* Department of Mathematcs, Unversty
More informationConstruction and number of self-dual skew codes over F _p 2
Constructon and number of self-dual skew codes over F _p 2 Delphne Boucher To cte ths verson: Delphne Boucher. Constructon and number of self-dual skew codes over F _p 2. Advances n Mathematcs of Communcatons,
More informationInner Product. Euclidean Space. Orthonormal Basis. Orthogonal
Inner Product Defnton 1 () A Eucldean space s a fnte-dmensonal vector space over the reals R, wth an nner product,. Defnton 2 (Inner Product) An nner product, on a real vector space X s a symmetrc, blnear,
More informationP.P. PROPERTIES OF GROUP RINGS. Libo Zan and Jianlong Chen
Internatonal Electronc Journal of Algebra Volume 3 2008 7-24 P.P. PROPERTIES OF GROUP RINGS Lbo Zan and Janlong Chen Receved: May 2007; Revsed: 24 October 2007 Communcated by John Clark Abstract. A rng
More informationChapter 5. Solution of System of Linear Equations. Module No. 6. Solution of Inconsistent and Ill Conditioned Systems
Numercal Analyss by Dr. Anta Pal Assstant Professor Department of Mathematcs Natonal Insttute of Technology Durgapur Durgapur-713209 emal: anta.bue@gmal.com 1 . Chapter 5 Soluton of System of Lnear Equatons
More informationPolynomials. 1 More properties of polynomials
Polynomals 1 More propertes of polynomals Recall that, for R a commutatve rng wth unty (as wth all rngs n ths course unless otherwse noted), we defne R[x] to be the set of expressons n =0 a x, where a
More informationIRREDUCIBLE POLYNOMIALS OF MAXIMUM WEIGHT OMRAN AHMADI AND ALFRED MENEZES
IRREDUCIBLE POLYNOMIALS OF MAXIMUM WEIGHT OMRAN AHMADI AND ALFRED MENEZES Abstract. We establsh some necessary condtons for the exstence of rreducble polynomals of degree n and weght n over F 2. Such polynomals
More informationCharacterizing the properties of specific binomial coefficients in congruence relations
Eastern Mchgan Unversty DgtalCommons@EMU Master's Theses and Doctoral Dssertatons Master's Theses, and Doctoral Dssertatons, and Graduate Capstone Projects 7-15-2015 Characterzng the propertes of specfc
More informationHyper-Sums of Powers of Integers and the Akiyama-Tanigawa Matrix
6 Journal of Integer Sequences, Vol 8 (00), Artcle 0 Hyper-Sums of Powers of Integers and the Ayama-Tangawa Matrx Yoshnar Inaba Toba Senor Hgh School Nshujo, Mnam-u Kyoto 60-89 Japan nava@yoto-benejp Abstract
More informationPRIMES 2015 reading project: Problem set #3
PRIMES 2015 readng project: Problem set #3 page 1 PRIMES 2015 readng project: Problem set #3 posted 31 May 2015, to be submtted around 15 June 2015 Darj Grnberg The purpose of ths problem set s to replace
More informationAmusing Properties of Odd Numbers Derived From Valuated Binary Tree
IOSR Journal of Mathematcs (IOSR-JM) e-iss: 78-578, p-iss: 19-765X. Volume 1, Issue 6 Ver. V (ov. - Dec.016), PP 5-57 www.osrjournals.org Amusng Propertes of Odd umbers Derved From Valuated Bnary Tree
More informationOn quasiperfect numbers
Notes on Number Theory and Dscrete Mathematcs Prnt ISSN 1310 5132, Onlne ISSN 2367 8275 Vol. 23, 2017, No. 3, 73 78 On quasperfect numbers V. Sva Rama Prasad 1 and C. Suntha 2 1 Nalla Malla Reddy Engneerng
More informationComplete subgraphs in multipartite graphs
Complete subgraphs n multpartte graphs FLORIAN PFENDER Unverstät Rostock, Insttut für Mathematk D-18057 Rostock, Germany Floran.Pfender@un-rostock.de Abstract Turán s Theorem states that every graph G
More informationSelf-complementing permutations of k-uniform hypergraphs
Dscrete Mathematcs Theoretcal Computer Scence DMTCS vol. 11:1, 2009, 117 124 Self-complementng permutatons of k-unform hypergraphs Artur Szymańsk A. Paweł Wojda Faculty of Appled Mathematcs, AGH Unversty
More informationThe Second Anti-Mathima on Game Theory
The Second Ant-Mathma on Game Theory Ath. Kehagas December 1 2006 1 Introducton In ths note we wll examne the noton of game equlbrum for three types of games 1. 2-player 2-acton zero-sum games 2. 2-player
More informationErrata to Invariant Theory with Applications January 28, 2017
Invarant Theory wth Applcatons Jan Drasma and Don Gjswjt http: //www.wn.tue.nl/~jdrasma/teachng/nvtheory0910/lecturenotes12.pdf verson of 7 December 2009 Errata and addenda by Darj Grnberg The followng
More informationCOMPLEX NUMBERS AND QUADRATIC EQUATIONS
COMPLEX NUMBERS AND QUADRATIC EQUATIONS INTRODUCTION We know that x 0 for all x R e the square of a real number (whether postve, negatve or ero) s non-negatve Hence the equatons x, x, x + 7 0 etc are not
More informationLinear, affine, and convex sets and hulls In the sequel, unless otherwise specified, X will denote a real vector space.
Lnear, affne, and convex sets and hulls In the sequel, unless otherwse specfed, X wll denote a real vector space. Lnes and segments. Gven two ponts x, y X, we defne xy = {x + t(y x) : t R} = {(1 t)x +
More informationA p-adic PERRON-FROBENIUS THEOREM
A p-adic PERRON-FROBENIUS THEOREM ROBERT COSTA AND PATRICK DYNES Advsor: Clayton Petsche Oregon State Unversty Abstract We prove a result for square matrces over the p-adc numbers akn to the Perron-Frobenus
More informationGELFAND-TSETLIN BASIS FOR THE REPRESENTATIONS OF gl n
GELFAND-TSETLIN BASIS FOR THE REPRESENTATIONS OF gl n KANG LU FINITE DIMENSIONAL REPRESENTATIONS OF gl n Let e j,, j =,, n denote the standard bass of the general lnear Le algebra gl n over the feld of
More informationp-adic Galois representations of G E with Char(E) = p > 0 and the ring R
p-adc Galos representatons of G E wth Char(E) = p > 0 and the rng R Gebhard Böckle December 11, 2008 1 A short revew Let E be a feld of characterstc p > 0 and denote by σ : E E the absolute Frobenus endomorphsm
More informationHMMT February 2016 February 20, 2016
HMMT February 016 February 0, 016 Combnatorcs 1. For postve ntegers n, let S n be the set of ntegers x such that n dstnct lnes, no three concurrent, can dvde a plane nto x regons (for example, S = {3,
More informationLecture 4: Universal Hash Functions/Streaming Cont d
CSE 5: Desgn and Analyss of Algorthms I Sprng 06 Lecture 4: Unversal Hash Functons/Streamng Cont d Lecturer: Shayan Oves Gharan Aprl 6th Scrbe: Jacob Schreber Dsclamer: These notes have not been subjected
More informationThe Jacobsthal and Jacobsthal-Lucas Numbers via Square Roots of Matrices
Internatonal Mathematcal Forum, Vol 11, 2016, no 11, 513-520 HIKARI Ltd, wwwm-hkarcom http://dxdoorg/1012988/mf20166442 The Jacobsthal and Jacobsthal-Lucas Numbers va Square Roots of Matrces Saadet Arslan
More informationOn the correction of the h-index for career length
1 On the correcton of the h-ndex for career length by L. Egghe Unverstet Hasselt (UHasselt), Campus Depenbeek, Agoralaan, B-3590 Depenbeek, Belgum 1 and Unverstet Antwerpen (UA), IBW, Stadscampus, Venusstraat
More informationOn the Nilpotent Length of Polycyclic Groups
JOURNAL OF ALGEBRA 203, 125133 1998 ARTICLE NO. JA977321 On the Nlpotent Length of Polycyclc Groups Gerard Endmon* C.M.I., Unerste de Proence, UMR-CNRS 6632, 39, rue F. Jolot-Cure, 13453 Marselle Cedex
More informationWeek 2. This week, we covered operations on sets and cardinality.
Week 2 Ths week, we covered operatons on sets and cardnalty. Defnton 0.1 (Correspondence). A correspondence between two sets A and B s a set S contaned n A B = {(a, b) a A, b B}. A correspondence from
More informationCharacter Degrees of Extensions of PSL 2 (q) and SL 2 (q)
Character Degrees of Extensons of PSL (q) and SL (q) Donald L. Whte Department of Mathematcal Scences Kent State Unversty, Kent, Oho 444 E-mal: whte@math.kent.edu July 7, 01 Abstract Denote by S the projectve
More informationOn the size of quotient of two subsets of positive integers.
arxv:1706.04101v1 [math.nt] 13 Jun 2017 On the sze of quotent of two subsets of postve ntegers. Yur Shtenkov Abstract We obtan non-trval lower bound for the set A/A, where A s a subset of the nterval [1,
More informationLecture 10 Support Vector Machines II
Lecture 10 Support Vector Machnes II 22 February 2016 Taylor B. Arnold Yale Statstcs STAT 365/665 1/28 Notes: Problem 3 s posted and due ths upcomng Frday There was an early bug n the fake-test data; fxed
More information18.781: Solution to Practice Questions for Final Exam
18.781: Soluton to Practce Questons for Fnal Exam 1. Fnd three solutons n postve ntegers of x 6y = 1 by frst calculatng the contnued fracton expanson of 6. Soluton: We have 1 6=[, ] 6 6+ =[, ] 1 =[,, ]=[,,
More informationSome Consequences. Example of Extended Euclidean Algorithm. The Fundamental Theorem of Arithmetic, II. Characterizing the GCD and LCM
Example of Extended Eucldean Algorthm Recall that gcd(84, 33) = gcd(33, 18) = gcd(18, 15) = gcd(15, 3) = gcd(3, 0) = 3 We work backwards to wrte 3 as a lnear combnaton of 84 and 33: 3 = 18 15 [Now 3 s
More informationChowla s Problem on the Non-Vanishing of Certain Infinite Series and Related Questions
Proc. Int. Conf. Number Theory and Dscrete Geometry No. 4, 2007, pp. 7 79. Chowla s Problem on the Non-Vanshng of Certan Infnte Seres and Related Questons N. Saradha School of Mathematcs, Tata Insttute
More information28 Finitely Generated Abelian Groups
8 Fntely Generated Abelan Groups In ths last paragraph of Chapter, we determne the structure of fntely generated abelan groups A complete classfcaton of such groups s gven Complete classfcaton theorems
More informationMTH 819 Algebra I S13. Homework 1/ Solutions. 1 if p n b and p n+1 b 0 otherwise ) = 0 if p q or n m. W i = rw i
MTH 819 Algebra I S13 Homework 1/ Solutons Defnton A. Let R be PID and V a untary R-module. Let p be a prme n R and n Z +. Then d p,n (V) = dm R/Rp p n 1 Ann V (p n )/p n Ann V (p n+1 ) Note here that
More informationDirichlet s Theorem In Arithmetic Progressions
Drchlet s Theorem In Arthmetc Progressons Parsa Kavkan Hang Wang The Unversty of Adelade February 26, 205 Abstract The am of ths paper s to ntroduce and prove Drchlet s theorem n arthmetc progressons,
More information