ÁÐÁÍÔÇÓÅÉÓ ÄÏÈÅÍÔÙÍ ÈÅÌÁÔÙÍ

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1 ÐáíåðéóôÞìéï ÊñÞôçò, ÔìÞìá Ìáèçìáôéêþí ëãåâñá (M 0 Åáñéíü ÅîÜìçíï 2009, ÅîÝôáóç Éïõíßïõ ÅîåôáóôÞò: ÄçìÞôñéïò ÍôáÞò ÁÐÁÍÔÇÓÅÉÓ ÄÏÈÅÍÔÙÍ ÈÅÌÁÔÙÍ ÈÅÌÁ ï Èåùñïýìå ôï óýíïëï ÈÝôïíôáò S := ( np λ j j λ,...,λ n Z. ( ε j,l :=, üôáí j = l, 0, üôáí j 6= l, ãéá êüèå j, l {,...,n}, Ý ïõìå ðñïöáíþò P l = n ε j,l j S, l {,...,n}. ÅÜí êüðïéïò åê ôùí,..., n åßíáé > 0, ôüôå S N 6=. Ùóôüóï, ôï üôé S N 6= åßíáé ðüíôïôå P áëçèýò, äéüôé áêüìç êáé åüí l < 0 ãéá êüèå l {,...,n}, Ý ïõìå l = n ( ε j,l j S N. Ùò åê ôïýôïõ, ôï S N äéáèýôåé åëü éóôï óôïé åßï, áò ðïýìå ôï d 0 = P n k j j. Èá áðïäåßîïõìå üôé d 0 = d. ÐñÜãìáôé ãéá ïéïäþðïôå óôïé åßï m = P n λjj ôïý S õðüñ åé Ýíá ìïíïóçìüíôùò ïñéóìýíï æåýãïò (q, r Z Z, ïýôùò þóôå íá éó ýåé m = qd 0 + r, üðïõ 0 r<d 0. ÕðïèÝôïíôáò üôé r>0 êáôáëþãïõìå óå êüôé ôï Üôïðï, êáèüóïí d 0 >r= n P (λ j k jq j S. ñá r =0= d 0 m êáé, åéäéêüôåñá, d 0 j ãéá êüèå j {,...,n}. ÅðéðñïóèÝôùò, ãéá ïéïíäþðïôå δ N, ãéá ôïí ïðïßï éó ýåé δ,...,δ n, Ý ïõìå [δ k,..., δ k n n]= δ d 0 = δ d 0, ïðüôå ôåëéêþò d 0 = d. ÈÅÌÁ 2ï '': óôù ôõ üí x HK. Ôüôå x = hk ãéá êüðïéá h H êáé k K. ÅðåéäÞ ôï HK åßíáé åî õðïèýóåùò õðïïìüäá ôþò G, Ý ïõìå x HK (âë (ii (c. ñá x = h 0 k 0 ãéá êüðïéá h 0 H êáé k 0 K, êáé x = x x =(h 0 k 0 =(k 0 (h 0 k 0 K (k 0 K êáé h 0 H (h 0 H x =(k 0 (h 0 KH. Ôïýôï óçìáßíåé üôé HK KH. Ãéá ôçí áðüäåéîç ôïý áíôéóôñüöïõ åãêëåéóìïý èåùñïýìå ôõ üí y KH. Ðñïöáíþò, y = kh ãéá êüðïéá k K êáé h H, êáé h H h H êáé k K k K y =(kh y HK. = h k ÅðåéäÞ ôï HK õðåôýèç üôé åßíáé õðïïìüäá ôþò G, Ý ïõìå y = y HK. ñá éó ýåé êáé áíôßóôñïöïò åãêëåéóìüò HK KH.

2 '': ÅðåéäÞ ôá H êáé K åßíáé õðïïìüäåò ôþò G, Ý ïõìå e G H êáé e G K, ïðüôå e G e G = e G HK. Åí óõíå åßá èåùñïýìå ôõ üíôá óôïé åßá x,x 2 HK. Åî ïñéóìïý õðüñ ïõí h,h 2 H êáé k,k 2 K, ôýôïéá þóôå íá éó ýïõí ïé éóüôçôåò x = h k êáé x 2 = h 2 k 2. ÅðéðñïóèÝôùò, ÊáôÜ óõíýðåéáí, k h 2 KH = HK h 3 H êáé k 3 K : k h 2 = h 3k 3. x x 2 = (h k (h 2 k 2 =h (k h 2 k 2 = h (h 3 k 3 k 2 =(h h {z 3 (k } 3 k 2 HK. {z} H K ÔÝëïò, ãéá ïéïäþðïôå x HK õðüñ ïõí h H êáé k K, ôýôïéá þóôå íá éó ýåé ç éóüôçôá x = hk, ïðüôå x =(hk = k h KH = HK. Óýìöùíá ìå ãíùóôþ ðñüôáóç (3.2.5 (ii ôï óýíïëï HK åßíáé õðïïìüäá ôþò G. ÈÅÌÁ 3ï Èåþñçìá ôïý Cyley. ÊÜèå ïìüäá (G, åßíáé éóüìïñöç ìå ìéá õðïïìüäá ôþò ïìüäáò (S G,. Áðüäåéîç. óôù (G, ôõ ïýóá ïìüäá (ìå ïõäýôåñü ôçò óôïé åßï ôï e G. Óå êüèå óôïé åßï g ôþò G áíôéóôïé ïýìå ìéá ìåôüôáîç L g ïñéæüìåíç ùò åîþò: L g : G G, L g (x =gx. (Ç áðåéêüíéóç L g åßíáé åíñéðôéêþ, äéüôé L g (x =L g (y = gx = gy = g gx = g gy = e Gx = e Gy = x = y, áëëü êáé åðéññéðôéêþ, äéüôé åüí z G, ôüôå L g g z = gg z = e G z = z. H L g ïíïìüæåôáé åî áñéóôåñþí ìåôáöïñü ìýóù ôïý g. óôù ôþñá G 0 ôï õðïóýíïëï { L g g G} ôþò S G. ÇðñÜîçìåôçí ïðïßá åßíáé åöïäéáóìýíç ç S G åßíáé ç óýíèåóç áðåéêïíßóåùí. Ùò åê ôïýôïõ, Ý ïõìå (L g L h (x =L g (L h (x = L g (hx =ghx = L gh (x, x G. ÊáôÜ óõíýðåéáí, ôï ãéíüìåíï äõï ôõ üíôùí óôïé åßùí ôïý G 0 áíþêåé óôï G 0. Ôï ôáõôïôéêü óôïé åßï id G ôþò S G áíþêåé óôï G 0 äéüôé éóïýôáé ìå ôçí L eg, åíþôïáíôßóôñïöïôþòl g åíôüò ôþò S G éóïýôáé ìå ôçí L g êáé áíþêåé êáé áõôü óôï G 0. Ôïýôï óçìáßíåé üôé ôï óýíïëï G 0 áðïôåëåß ìéá õðïïìüäá ôþò S G äõíüìåé ãíùóôþò ðñïôüóåùò (âë (ii. Ç áðåéêüíéóç G G 0, g 7 L g, åßíáé ðñïöáíþò åðéññéðôéêþ êáé ìåôáöýñåé ôïí ðïëëáðëáóéáóìü ôþò G óôç óýíèåóç áðåéêïíßóåùí ôþò G 0 (gh 7 L gh = L g L h. ÅîÜëëïõ, ç åí ëüãù áðåéêüíéóç åßíáé êáé åíñéðôéêþ, áöïý áðü ôçí L g = L h Ýðåôáé üôé g = L g (e G =L h (e G =h. Êáô' áõôüí ôïí ôñüðï êáôáóêåõüóáìå Ýíáí éóïìïñöéóìü ìåôáîý ôþò G êáé ôþò õðïïìüäáò G 0 ôþò ïìüäáò S G. ÈÅÌÁ 4ï ï Èåþñçìá Éóïìïñöéóìþí Äáêôõëßùí. óôù f : R S Ýíáò ïìïìïñöéóìüò äáêôõëßùí. Ôüôå R/Ker(f = Im (f =f (R. 3ï Èåþñçìá Éóïìïñöéóìþí Äáêôõëßùí. ÅÜí ï R åßíáé Ýíáò äáêôýëéïò êáé ôá I,J éäåþäç ôïý R ìå I J, ôüôå Ý ïõìå Áðïäåéîç. óôù f ç áðåéêüíéóç R/J = (R/I /(J/I. f : R (R/I /(J/I, 7 ( + I+(J/I, R. 2

3 ÅðåéäÞ f = π 2 π,üðïõπ : R (R/I êáé π 2 : R/I (R/I /(J/I ïé öõóéêïß åðéìïñöéóìïß, ç f åßíáéýíáò(êáëþòïñéóìýíïòåðéìïñöéóìüòäáêôõëßùí. Óýìöùíá ìå ôï ï èåþñçìá éóïìïñöéóìþí, R/Ker (f = (R/I /(J/I. ¼ìùò Ker(f = ª R f( =0 (R/I/(J/I = ª R π 2 (π ( = 0 (R/I/(J/I = ª R π 2( + I =0 (R/I/(J/I = { R + I Ker (π 2} = { R + I (J/I} = J, áð' üðïõ Ýðåôáé ôï æçôïýìåíï. ÈÅÌÁ 5ï (i ÅÜí f(x = P n j=0 jxj Z[X]r{0 Z[X] } ìå n 6= 0, ôüôå ï áñéèìüò cont(f(x := ìêä( 0,,..., n êáëåßôáé ðåñéå üìåíï ôïý ðïëõùíýìïõ f(x. ÊÜèå ôýôïéïõ åßäïõò ðïëõþíõìï ìå cont(f(x = êáëåßôáé ðñùôáñ éêü ðïëõþíõìï. ËÞììá ôïý Guss. Ôï ãéíüìåíï äõï ðñùôáñ éêþí ðïëõùíýìùí (áíçêüíôùí óôï Z[X]r{0 Z[X] } åßíáé ðüíôïôå Ýíá ðñùôáñ éêü ðïëõþíõìï. Áðüäåéîç. óôù üôé ôá f(x,g(x Z[X]r{0 Z[X] } åßíáé äõï ðñùôáñ éêü ðïëõþíõìá. Áò õðïèýóïõìå üôé ôï ãéíüìåíü ôïõò f(xg(x äåí åßíáé ðñùôáñ éêü ðïëõþíõìï. óôù p Ýíáò ðñþôïò áñéèìüò ðïõ äéáéñåß ôï cont(f(xg(x êáé Ýóôù Ψ p : Z[X] Z p [X] ï ïìïìïñöéóìüò äáêôõëßùí ï ïñéæüìåíïò ìýóù ôïý ôýðïõ P Ψ p ( n j X j P := n [ j ] p X j P, n j X j Z[X]. j=0 j=0 Ôüôå Ý ïõìå ðñïöáíþò Ψ p (f(x, Ψ p (g(x Z p [X] êáé j=0 Ψ p(f(xψ p(g(x = Ψ p(f(xg(x = 0 Zp [X], ïðüôå åßôå ï p äéáéñåß êüèå óõíôåëåóôþ ôïý ðïëõùíýìïõ f(x åßôå ï p äéáéñåß êüèå óõíôåëåóôþ ôïý ðïëõùíýìïõ g(x. Áõôü óçìáßíåé üôé åßôå ôï f(x åßôå ôï g(x äåí åßíáé ðñùôáñ éêü. ôïðï! (ii Ðñüôáóç. ÅÜí Ýíá ðïëõþíõìï f(x Z[X] åßíáé áíüãùãï åíôüò ôïý ðïëõùíõìéêïý äáêôõëßïõ Z[X], ôüôå åßíáé áíüãùãï êáé åíôüò ôïý Q[X]. Áðüäåéîç. Áò õðïèýóïõìå üôé ôï f(x Z[X] äåí áíüãùãï åíôüò ôïý Q[X]. Ôüôå õðüñ ïõí ðïëõþíõìá g(x,h(x Q[X] âáèìïý, ôýôïéá þóôå íá éó ýåé ç éóüôçôá f(x =g(xh(x. Äß ùò âëüâç ôþò ãåíéêüôçôáò ìðïñïýìå íá õðïèýóïõìå üôé ôï f(x åßíáé ðñùôáñ éêü (åéäüëëùò èåùñïýìå áíô' áõôïý ôï f(x. óôù (êáé áíôéóôïß ùò, b ôï åêð ôùí ðáñïíïìáóôþí ôùí óõíôåëåóôþí ôïý g(x (êáé cont(f(x áíôéóôïß ùò, ôïý h(x. Ôüôå g(x,bh(x Z[X] b f(x =(g(x(bh(x Z[X]. ÈÝôïíôáò c := cont(g(x êáé c 2 := cont(bh(x Ý ïõìå g(x =c eg(x, bh(x =c 2 e h(x, 3

4 ãéá êüðïéá eg(x, e h(x Z[X] âáèìïý ìå cont(eg(x = cont( e h(x =. Ðñïöáíþò, b f(x =c c 2 eg(x e h(x, cont(f(x = = cont(b f(x = b, cont(eg(x e h(x (Guss = = cont(c c 2 eg(x e h(x = c c 2, = b = cc2, ïðüôå f(t =eg(t e h(t Z[t], ðñüãìá ðïõ óçìáßíåé üôé ôï f(t äåí áíüãùãï ïýôå åíôüò ôïý Z[t]. ÈÅÌÁ 6ï (i ÅðåéäÞ ìêä(m, n =, ôï èåþñçìá ôïý Euler ðåñß éóïôéìéþí äßäåé m ϕ(n (mod n ³ mn m ϕ(n (n ϕ(m, n ϕ(m (mod m ïðüôå mn m ϕ(n n ϕ(m (m ϕ(n +n ϕ(m mn m ϕ(n +n ϕ(m m ϕ(n +n ϕ(m (modmn. (ii ÅðåéäÞ ìêä(5, 24 =, çãñáììéêþéóïôéìßá5x 3(mod 24 äéáèýôåé áêñéâþò ìßá ëýóç x 0 êáôü ìüäéï m (âë ÃñÜöïíôáò 24 = 2 3 3, ìýóù ãíùóôïý ôýðïõ (âë ðñïóäéïñßæïõìå ôçí ôéìþ ϕ (24 = ( (3 = 8. ÅðåéäÞ (êáôü ôï èåþñçìá ôïý Euler ðåñß éóïôéìéþí Ý ïõìå 5 ϕ(24 (mod 24, ìðïñïýìå íá èýóïõìå ùò x 0 := 5 ϕ(24 3=5 7 3 = Ùóôüóï,åßèéóôáéíá ðáñý ïõìåôçæçôïýìåíçëýóçóôçíáíçãìýíç ôçò ìïñöþ. ÅðåéäÞ ëïéðüí = , Ý ïõìå [x 0 ] 24 =[5] 24, ïðüôå 5 5 3(mod 24. (Åíáëëáêôéêþò, ãéá íá ìçí êáôáöýãïõìå óå äéáßñåóç ìå Ýíáí ôüóï ìåãüëï áñéèìü, Ý ïõìå ôç äõíáôüôçôá íá åðé åéñçìáôïëïãþóïõìå ùò åîþò: 5 2 (mod 24, ïðüôå 5 6 (mod (mod (mod 24. ÈÅÌÁ 7ï (i óôù H ìéá ðåðåñáóìýíùò ðáñáãüìåíç õðïïìüäá ôþò (Q, +. óôù üôé À H =, 2,..., r, d d 2 d r üðïõ r N, j Z, d j Zr{0}, j {,...,r}. ÅÜí d := d = d 2 d r d d. r d r = rd d r d d r Q d j, ôüôå = H À, d êáé ç H -ùòõðïïìüäáìéáòêõêëéêþòïìüäáò-ïöåßëåéíáåßíáéêõêëéêþ(âë (i. (ii Áò õðïèýóïõìå üôé ç (Q, + åßíáé ðåðåñáóìýíùò ðáñáãüìåíç. Ôüôå, âüóåé ôïý (i êáé ôïý üôé äåí åßíáé ôåôñéììýíç, èá éó ýåé Q = b ãéá êüðïéá, b Zr{0}. ÅðåéäÞ 2b Q èá ðñýðåé íá õðüñ åé n Z, ôýôïéï þóôå íá éó ýåé 2b = n b = n = / Z, 2 ðñüãìá Üôïðï. ÊáôÜ óõíýðåéáí, ç (Q, + äåí åßíáé ðåðåñáóìýíùò ðáñáãüìåíç. (iii ÅðåéäÞ ãéá êüèå öõóéêü áñéèìü i, =(i + H i! (i+! i+ êáé / H (i+! i, Ý ïõìå ðñïöáíþò H i $ H i+. Ãéá íá äåßîïõìå üôé Q = S n Hn, áñêåß íá äåßîïõìå üôé ãéá êüèå ñçôü áñéèìü Q b õðüñ åé i 0 ìå H b i 0. Ôïýôï üìùò Ýðåôáé áðü ôï üôé (èýôïíôáò ùò i 0 ôï b ëáìâüíïõìå b = sign(b b = sign(b =( b! sign(b b b! H b. (iv Ôï U ðåñéý åé ôï 0 êáé åßíáé êëåéóôü ùò ðñïò ôçí ðñüóèåóç (üðùòêáíåßòìðïñåßüìåóáíáäéáðéóôþóåé. ÅðéðñïóèÝôùò, åüí U, ôüôå êáé U, êáèüôé ï åßíáé Üñôéïò êáé ï b ðåñéôôüò áñéèìüò. b b Óõíåðþò ôï U óõãêñïôåß ìéá (ðñïöáíþò ìç ôåôñéììýíç õðïïìüäá ôþò (Q, +. ÌÜëéóôá,U $Q, áöïý ð.. QrU. (ÅÜí =, ìå Üñôéï êáé b ðåñéôôü, èá êáôáëþãáìå óå Üôïðï óõìðýñáóìá ôïý ôýðïõ: 3 3 b 4

5 Üñôéïò = ðåñéôôüò. ÁðïìÝíåé íá äåßîïõìå üôé ç U äåí åßíáé êõêëéêþ. ÅÜí õðþñ å x = U, 6= 0, ìå b U = hxi, ôüôå ð.. U (êáèüôé ôï 3b ðáñáìýíåé ðåñéôôüò, ïðüôå èá õðþñ å êüðïéïò n Z, ôýôïéïò 3b þóôå íá éó ýåé 3b = nx = n b = n = / Z, 3 ðñüãìá Üôïðï. ÊáôÜ óõíýðåéáí, ç U äåí åßíáé êõêëéêþ. ÈÅÌÁ 8ï (i Êáô' áñ Üò áðïäåéêíýïõìå üôé ôï êýíôñï Z(G :={ g G xg = gx, x G} ìéáò ïìüäáò G áðïôåëåß ìéá õðïïìüäá ôþò G. Ðñïöáíþò, e G Z(G. ÅðéðñïóèÝôùò, ãéá ôõ üíôá g, h Z(G êáé x G Ý ïõìå (ëüãù ôþò ðñïóåôáéñéóôéêþò éäéüôçôáò êáé ôïý ïñéóìïý ôïý Z(G êáé xh = hx h (xh = h h x = e G x = x h x = h (xh h = xh x(gh =(xgh =(gxh = g(xh =g(h x=(gh x gh Z(G. ñá ôï êýíôñï Z(G åßíáé üíôùò ìéá õðïïìüäá ôþò G (âë (iii. Åí óõíå åßá, áðïäåéêíýïõìå üôé áõôü ðñüêåéôáé ãéá ìéá ïñèüèåôç õðïïìüäá ôþò G. Ðñïò ôïýôï áñêåß íá èåùñçèïýí ôõ üíôá óôïé åßá g Z(G êáé x G êáé íá ëçöèåß õð' üøéí üôé xg = gx xgx = gxx = g xz(gx Z(G. (ii ÅÜí ç ðçëéêïïìüäá G/Z(G åßíáé êõêëéêþ, ôüôå õðüñ åé êüðïéï g G, ôýôïéï þóôå íá éó ýåé G/Z(G =hgz(gi. Èåùñïýìå ôõ üíôá óôïé åßá, b G. Ôüôå õðüñ ïõí m, n Z : Z(G =(gz(g m = g m Z(G bz(g =(gz(g n h,h = g n 2 Z(G : = g m h,b= g n h 2. Z(G ñá b =(g m h (g n h 2 =g m (h g n h 2 = g m (g n h h 2 = g m+n h h 2 b =(g n h 2(g m h =g n (h 2g m h = g n (g m h 2 h = g m+n h 2h h,h 2 Z(G h h 2 = h 2 h b = b êáé ç G åßíáé áâåëéáíþ (êáé, ùò åê ôïýôïõ, ç G/Z(G åßíáé ôåôñéììýíç. (iii Ç ïìüäá ôùí ôåôñáíßùí,þôïéçõðïïìüäáq ôþò SU 2 (C ç ðáñáãüìåíç áðü ôïõò ðßíáêåò j êáé k, Ý åé ôüîç 8, êáèüôé Q = {±I 2, ±i, ±j, ±k}, (i := jk. ÓçìåéùôÝïí üôé ç Q äåí åßíáé áâåëéáíþ (áöïý ð.. kj 6= jk, ïðüôå Z(Q $ Q. ÅðéðñïóèÝôùò, åêôüò ôïý I 2 êáé ôï I 2 áíþêåé óôï êýíôñï ôþò Q,äéüôé ãéá êüèå ðßíáêá A áíþêïíôá óôçí Q Ý ïõìå A ( I 2=( A I 2 = I 2 ( A =( I 2 A. Óõíåðþò, {±I 2} Z(Q. (Ôï üôé ôï êýíôñï Z(Q ôþò Q åßíáé ìç ôåôñéììýíç õðïïìüäá ôþò Q ìðïñåß íá áðïäåé èåß êáé åíáëëáêôéêþò êüíïíôáò ñþóç ôïý èåùñþìáôïò 3.7.8, áöïý Q =2 3. ÅðåéäÞ ëïéðüí ôï êýíôñï Z(Q ôþò Q åßíáé ìéá ìç ôåôñéììýíç, ãíþóéá êáé ïñèüèåôç õðïïìüäá ôþò Q, ôï èåþñçìá ôïý Lgrnge ìüò ðëçñïöïñåß üôé Z(Q {2, 4}, ïðüôå Q/Z(Q = Q Z(Q {2, 4}. Ùò ãíùóôüí, êüèå ïìüäá ôüîåùò 2 åßíáé êõêëéêþ êáé éóüìïñöç ôþò (Z 2, + (âë. ðüñéóìá êáé èåþñçìá (ii. ÊáôÜ óõíýðåéáí, åüí õðïèýôáìå üôé Q/Z(Q =2, èá åß áìå Q/Z(Q = Z 2 êáé âüóåé ôïý (ii èá óõìðåñáßíáìå üôé ç ßäéá ç Q åßíáé áâåëéáíþ, Þôïé êüôé ôï åóöáëìýíï. Ùò åê ôïýôïõ, Q/Z(Q =4 Z(Q =2 Z(Q = Z 2. (ÌÜëéóôá, âüóåé ôùí ðñïáíáöåñèýíôùí, Z(Q ={±I 2 }. 5

6 (iv Åñãáæüìáóôå ìå «åéò Üôïðïí áðáãùãþ». ÕðïèÝôïõìå üôé õðüñ åé êüðïéï óôïé åßï σ S n r{id}, ôýôïéï þóôå íá éó ýåé σ τ = τ σ Þ-éóïäõíÜìùò-σ τ σ = τ, ãéá êüèå τ S n. Ôï σ (óýìöùíá ìå ãíùóôü èåþñçìá, âë ãñüöåôáé õðï ôç ìïñöþ åðáëëþëùí óõíèýóåùí (ðåðåñáóìýíïõ ðëþèïõò áíü äýï îýíùí ìåôáîý ôïõò êýêëùí ìþêïõò 2, áò ðïýìå σ = c c 2 c ν. óôù üôé c =[ 2 k ], ãéá êüðïéïí k N, 2 k n. ÅîåôÜæïõìå äýï ðåñéðôþóåéò ùñéóôü: Ðåñßðôùóç ðñþôç. ÅÜík 3, ôüôå èåùñþíôáò ùò τ ôïí 2-êýêëï [ 2] êáôáëþãïõìå óå Üôïðï, êáèüóïí σ τ σ = σ [ 2 ] σ =[σ( σ( 2 ] = [ 2 3 ] 6= [ 2 ]=τ. Ðåñßðôùóç äåýôåñç. ÅÜí k =2, ôüôå èåùñþíôáò ùò τ ôïí 3-êýêëï [ 2 3] êáôáëþãïõìå åê íýïõ óå Üôïðï, êáèüóïí σ τ σ = σ [ 2 3] σ =[σ( σ( 2 σ ( 3] = [ 2 σ ( 3], üðïõ σ ( 3 / {, 2 } êáé σ τ σ ( 2 = 6= 3 = τ( 2. ñá ôåëéêþò Z(S n = {id} ãéá êüèå öõóéêü áñéèìü n 3. ÈÅÌÁ 9ï óôù üôé ôá K êáé L åßíáé äõï óþìáôá, ç áñáêôçñéóôéêþ ôùí ïðïßùí äåí éóïýôáé ïýôå ìå 2 ïýôå ìå 3. ÅÜí ç f : K L åßíáé ìéá áðåéêüíéóç ðïõ ðëçñïß ôéò óõíèþêåò f(x + y =f(x+f(y, x, y K, f( K = L, f(x 3 =f(x 3, x K, æçôåßôáé íá áðïäåé èåß üôé ç f åßíáé Ýíáò ïìïìïñöéóìüò óùìüôùí, äçëáäþ üôé Áðïäåéîç. ÂÞìá ï. Êáô' áñ Üò éó ýåé ç éóüôçôá f(xy =f(xf(y, x, y K. f(x 2 =f(x 2, x K. ( ÐñÜãìáôé ëüãù ôþò ôñßôçò åê ôùí áíùôýñù óõíèçêþí Ý ïõìå ãéá êüèå x K: f((x + K 3 =f(x + K 3. ÌÝóù ôùí áíùôýñù óõíèçêþí ôï áñéóôåñü ìýëïò ôþò ôåëåõôáßáò éóüôçôáò ãñüöåôáé ùò f(x 3 +3x 2 +3x + K=f(x 3 +3f(x 2 +3f(x+f( K=f(x 3 +3f(x 2 +3f(x+ L. êáé ôï äåîéü ôçò ìýëïò ùò f(x + K 3 =(f(x+f( K 3 =(f(x+ L 3 = f(x 3 +3f(x 2 +3f(x+ L. ÊáôÜ óõíýðåéáí, 3 f(x 2 f(x 2 =( L + L + L f(x 2 f(x 2 =0 L áñ(l 6= 3 L + L + L 6=0 L f(x 2 =f(x 2. ÂÞìá 2ï. Èåùñïýìå ôõ üíôá óôïé åßá x, y K. ÅðåéäÞ 2xy =(x + y 2 x 2 y 2, ýóôåñá áðü åöáñìïãþ ôþò f óå áìöüôåñá ôá ìýëç áõôþò ôþò éóüôçôáò ç ( äßäåé 2f(xy = f((x + y 2 f(x 2 f(y 2 =f(x + y 2 f(x 2 f(y 2 = (f(x+f(y 2 f(x 2 f(y 2 =2f(xf(y ïðüôå 2(f(xy f(xf(y = ( L + L (f(xy f(xf(y = 0 L áñ(l 6= 2 L + L 6= 0 L f(xy =f(xf(y êáé ç f åßíáé üíôùò Ýíáò ïìïìïñöéóìüò óùìüôùí. 6

7 ÈÅÌÁ 0ï (i Äßäåôáé ôï ðïëõþíõìï g(x =X 4 +5X 3 9X 2 4X +24 Z[X]. óôù r Q,ôÝôïéïò þóôå íá éó ýåé çéóüôçôág(r =0. Éó õñéóìüò: r = 6. Ðñùôç áðïäåéîç éó õñéóìïý. ÅêöñÜæïíôáòôïír õðü ôç ìïñöþ r = λ, üðïõ λ Z, μ Zr{0} êáé μ ìêä(λ, μ =, ôï êñéôþñéï ñçôþí èýóåùí ìçäåíéóìïý 5.5. ìáò ðëçñïöïñåß üôé λ 24 êáé μ, ïðüôå λ {±, ±2, ±3, ±4, ±6, ±8, ±2, ±24},μ {±} r {±, ±2, ±3, ±4, ±6, ±8, ±2, ±24}. Åêôåëþíôáò áðåõèåßáò ðñüîåéò äéáðéóôþíïõìå üôé g( = 7 6= 0, g( = 25 6= 0, g(2 = 6 6= 0, g( 2 = 8 6= 0, g(3 = 7 6= 0, g( 3 = 69 6= 0, g(4 = 400 6= 0, g( 4 = 28 6= 0, g(6 = 992 6= 0, g( 6 = 0, g(8 = = 0, g( 8 = 096 6= 0, g(2 = = 0, g( 2 = = 0, g(24 = = 0, g( 24 = = 0. Äåõôåñç áðïäåéîç éó õñéóìïý. Ãéá íá áðïöýãïõìå ôéò ðïëëýò ðñüîåéò åðé åéñçìáôïëïãïýìå ùò åîþò: Ðñïöáíþò, r Z, äéüôé ôï g(x åßíáé ìïíéêü. ÊáôÜ ôï (ii ôþò ðñïôüóåùò 5.3.2, g(r =0 X r g(x h(x Z[X] :g(x =(X r h(x, ïðüôå g( = 7 = ( rh( r 7 r {±, ±7}. ñá r {0, 2, 6, 8}. Áñêåß íá ðáñáôçñþóïõìå üôé g(0 = 24 6= 0, g(2 = 6 6= 0, g( 6 = 0, g(8 = = 0 (ii ÄïèÝíôùí äõï áêåñáßùí áñéèìþí, b, èåùñïýìå ôï ðïëõþíõìï f(x =X 4 + X 2 + b 2 Z[X]. Æçôåßôáé íá áðïäåé èåß üôé ôï f(x äåí åßíáé áíüãùãï åíôüò ôïý Q[X] åüíêáéìüíïíåüíôïõëü éóôïí Ýíáò åê ôùí 2 4b 2, 2b, 2b éóïýôáé ìå ôï ôåôñüãùíï êüðïéïõ áêåñáßïõ áñéèìïý. Áðïäåéîç. '' ÅÜí c Z : 2 4b 2 = c 2, ôüôå ³ f(x = X 2 2 ÅÜí c Z :2b = c 2, ôüôå ÔÝëïò, åüí c Z : 2b = c 2, ôüôå 2 ³ c 2 2 = ³ X 2 ³ c ³ X 2 2 f(x = X 2 + cx + b X 2 cx + b. f(x = X 2 + cx b X 2 cx b. ³ + c. 2 '' ÕðïèÝôïõìå üôé ôï f(x äåí åßíáé áíüãùãï åíôüò ôïý Q[X]. Ôï f(x èá äéáèýôåé êüðïéï ðïëõþíõìï (áíþêïí óôïí Q[X] âáèìïý Þ 2 ùò ðáñüãïíôü ôïõ. ÅÜí ôï ðïëõþíõìï áõôü åßíáé âáèìïý, ôüôå ôï f(x Ý åé ðñïöáíþò ìéá èýóç ìçäåíéóìïý r Q. ÅÜí r =0, ôüôå b =0, ïðüôå ôï 2 4b 2 = 2 åßíáé ôï ôåôñüãùíï ôïý áêåñáßïõ áñéèìïý. ÅÜí r 6= 0, ôüôå r 6= r êáé f(r =f( r =0, ïðüôå (óýìöùíá ìå ôá (ii êáé X r f(x X + r f(x (X r(x + r =X 2 r 2 f(x. Áõôü óçìáßíåé üôé üôáí ôï f(x Ý åé Ýíáí ðñùôïâüèìéï ðáñüãïíôá (åíôüò ôïý Q[X] êáé X - f(x, ôüôå ôï f(x Ý åé êáô' áíüãêçí êáé êüðïéï äåõôåñïâüèìéï ðïëõþíõìï ùò ðáñüãïíôü ôïõ (åíôüò ôïý Q[X]. Ùò åê ôïýôïõ, ìðïñïýìå áðü ôïýäå êáé óôï åîþò (äß ùò âëüâç ôþò ãåíéêüôçôáò íá õðïèýóïõìå (êáôü ôçí 7

8 õðïëåéðüìåíç áðïäåéêôéêþ ðïñåßá üôé ôï f(x Ý åé êüðïéï äåõôåñïâüèìéï ðïëõþíõìï ùò ðáñüãïíôü ôïõ (åíôüò ôïý Q[X], Þôïé üôé g(x,h(x Q[X] : f(x =g(xh(x, ìå deg(g(x = deg(h(x = 2. óôù κ (êáé áíôéóôïß ùò, λ ôï åêð ôùí ðáñïíïìáóôþí ôùí óõíôåëåóôþí ôïý g(x (êáé áíôéóôïß ùò, ôïý h(x. Ôüôå f(t =eg(t e h(t, eg(x, e h(x Z[X], üðïõ eg(x := κg(x, e h(x := λh(x (üðùò óôçí áðüäåéîç ôïý (ii ôïý èýìáôïò 5, ìå cont(κ g(x cont(λ h(x deg(eg(x = deg( e h(x = 2. ÅðåéäÞ ôï f(x åßíáé ìïíéêü, ôá eg(x, e h(x èá ãñüöïíôáé õðü ôçí ìïñöþ eg(x =X 2 + sx + t, e h(x =X 2 + ux + v, üðïõ s, t, u, v åßíáé êáôüëëçëïé áêýñáéïé, ïðüôå ôï f(x èá ðáñáãïíôïðïéåßôáé åí ôýëåé åíôüò ôïý Z[X] ùò åîþò: f(x = X 2 + sx + t X 2 + ux + v. Åîéóþíïíôáò ôïõò óõíôåëåóôýò ôïý X 3 óôá äýï ìýëç ëáìâüíïõìå u = s. ÅðïìÝíùò, êáôüðéí åîéóþóåùò ôùí óõíôåëåóôþí ôùí, X, X 2 óôá äýï ìýëç óõìðåñáßíïõìå üôé vt = b 2, s(v t =0, t+ v = s 2 +. ÅÜí s =0, ôüôå 2 4b 2 =(t + v 2 4vt =(t v 2, Þôïé ôï 2 4b 2 éóïýôáé ìå ôï ôåôñüãùíï ôïý áêåñáßïõ áñéèìïý t v. ÅÜí s 6= 0, ôüôå v = t t 2 = b 2 t = ±b. Óôçí ðåñßðôùóç êáôü ôçí ïðïßá v = t = b Ý ïõìå 2b = s 2 (= ôï ôåôñüãùíï ôïý áêåñáßïõ s, åíþ óôçí ðåñßðôùóç êáôü ôçí ïðïßá v = t = b Ý ïõìå 2b = s 2 (= ôï ôåôñüãùíï ôïý áêåñáßïõ s ÔáèåùñçôéêÜèÝìáôá,2,3,4êáé5åß áíäéäá èåßêáôüôçäéüñêåéáôùíðáñáäüóåùí. Ôá (i êáé (ii ôïý èýìáôïò 6 åß áí äïèåß ùò áóêþóåéò êáé ëõèåß áðü ôïí âïçèü óôéò þñåò ôùí öñïíôéóôçñßùí (âë. áóêþóåéò êáé 3 (ii ôïý 4ïõ êáôáëüãïõ ðñïôåéíïìýíùí áóêþóåùí. Ôï èýìá 7 áðáéôïýóå ìüíïí ôç ãíþóç ôïý ôé åßíáé ìéá ðåðåñáóìýíùò ðáñáãüìåíç ïìüäá, ôïý üôé êüèå õðïïìüäá ìéáò êõêëéêþò ïìüäáò åßíáé êõêëéêþ, êáèþò êáé êüðïéá óôïé åéþäç áñéèìïèåùñçôéêü åðé åéñþìáôá. Óôï èýìá 0 õðåéóþñ ïíôï áñêåôýò ðñüîåéò êáé ðáñáãïíôïðïéþóåéò. Ùóôüóï, ôá áðáéôïýìåíá êñéôþñéá áíáãùãéìüôçôáò ðïëõùíýìùí (ìå óõíôåëåóôýò åéëçììýíïõò áðü ôïõò áêåñáßïõò êáé ôïõò ñçôïýò Þôáí ôá ðëýïí ñçóôéêü (êáé åí ðïëëïßò êïéíüôïðá. Ôá èýìáôá 8 êáé 9 Þôáí êüðùò ðéï áðáéôçôéêü. Ôï 8 ðñïûðýèåôå êáëþ ãíþóç ôïý êåöáëáßïõ ðåñß ïìüäùí êáé óõíäõáóìü áñêåôþí èåùñçôéêþí áðïôåëåóìüôùí, åíþ ôï 9 ðñïûðýèåôå Ýíáí êüðïéï âáèìü åõñçìáôéêüôçôáò (ãéá ôçí åöáñìïãþ ôïý «ëõôñùôéêïý» ôå íüóìáôïò ôïý ðñþôïõ âþìáôïò. 8