ÁÐÁÍÔÇÓÅÉÓ ÄÏÈÅÍÔÙÍ ÈÅÌÁÔÙÍ

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1 ÐáíåðéóôÞìéï ÊñÞôçò, ÔìÞìá Ìáèçìáôéêþí Èåùñßá Äáêôõëßùí êáé Modules (M ) ÅîÝôáóç Éïõíßïõ 010 ÅîåôáóôÞò: ÄçìÞôñéïò ÍôáÞò ÁÐÁÍÔÇÓÅÉÓ ÄÏÈÅÍÔÙÍ ÈÅÌÁÔÙÍ ÈÅÌÁ 1ï Âë. èåþñçìá.5.0 (óôéò óçìåéþóåéò). ÈÅÌÁ ï Âë. åäüöéá 3.3.1, 3.3., êáé ÈÅÌÁ 3ï Âë. èåþñçìá ÈÅÌÁ 4ï (i) Âë. ðñüôáóç.3.5. (ii) Âë. èåþñçìá ÈÅÌÁ 5ï Âë. èåþñçìá ÈÅÌÁ 6ï Áò õðïèýóïõìå üôé õðüñ åé êüðïéïò ìåôáèåôéêüò äáêôýëéïò R ìå ìïíáäéáßï óôïé åßï, ìå ôçí ðïëëáðëáóéáóôéêþ ïìüäá ôùí áíôéóôåøßìùí óôïé åßùí ôïõ (R, ) Ý ïõóá ôüîç R =5. ÅðåéäÞ ôï 5 åßíáé ðñþôïò áñéèìüò, êáôü ôï (iii) (ôþò äïèåßóáò óçìåéþóåùò) ç R åßíáé êõêëéêþ ïìüäá. Áò õðïèýóïõìå üôé ôï a R åßíáé Ýíáò ãåííþôïñáò áõôþò. Ðñïöáíþò, a 5 =1 R êáé a 6= 1 R (äéüôé ç (R, ) åßíáé ìç ôåôñéììýíç) êáé R = {1 R,a,a,a 3,a 4 }. Ùò ãíùóôüí, 0 R / R êáé {±1 R} R (âë. åäüöéï 1..9). ÅðåéäÞ ( 1 R ) =1 R, Ý ïõìå ord( 1 R ) {1, } (âüóåé ôïý ïñéóìïý ôþò ôüîåùò óôïé åßïõ êáé ôïý (i) ôþò äïèåßóáò óçìåéþóåùò). ÅðåéäÞ ord( 1 R) 5 (âë. ôï (ii) ôþò äïèåßóáò óçìåéþóåùò), Ý ïõìå êáô' áíüãêçí ord( 1 R )=1, ïðüôå 1 R =1 R (= ôï ïõäýôåñï óôïé åßï ôþò (R, )). Áõôü óçìáßíåé üôé 1 R =0 R, áð' üðïõ Ýðåôáé üôé áñ(r) =(âë (iii) êáé ðñüôáóç 1.4.3). ÐñïêåéìÝíïõ íá êáôáëþîïõìå óå Üôïðï, áñêåß íá ðñïóäéïñßóïõìå êüðïéï óôïé åßï b R ôþò ìïñöþò b =1 R +c(a+1 R ),c R r{1}, ìå b k =1 R, ãéá êüðïéï k {, 3, 4}. ÐñÜãìáôé õðïèýôïíôáò üôé õðüñ åé Ýíá ôýôïéï b, èá éó ýåé ord(b) k (âë. ôï (i) ôþò äïèåßóáò óçìåéþóåùò) êáé, ùò åê ôïýôïõ, ord(b) {1,, 3, 4}. Çðåñßðôùóçüðïõord(b) 6= 1áðïêëåßåôáé, êáèüóïí ôïýôï èá áíôýöáóêå ðñïò ôï (ii) ôþò äïèåßóáò óçìåéþóåùò (áöïý - 5, 3-5 êáé 4-5). ñá ord(b) = 1 b =1 R c(a +1 R )=0 R a +1 R = c 1 (c(a +1 R )) = c 1 0 R =0 R a = 1 R =1 R, ðñüãìá Üôïðï, áöïýa 6= 1 R. Ï ðñïóäéïñéóìüò åíüò ôýôïéïõ b ãßíåôáé ìå äïêéìýòãéáôéòôéìýòôïýc, õøþóåéòóåêüðïéááðüôéòôñåéòåðéôñåðôýòäõíüìåéò êáé ñþóç ôïý ãåãïíüôïò üôé ï äáêôýëéüò ìáò Ý åé áñáêôçñéóôéêþ. Åðß ðáñáäåßãìáôé, ïñßæïíôáò ùò b ôï b := 1 R + a (a +1 R) (ìå c := a R r{1}, êáé ùñßò íá ãíùñßæïõìå åê ôùí ðñïôýñùí üôé ôï åí ëüãù b áíþêåé óôçí R ), ëáìâüíïõìå (ýóôåñá áðü ýøùóç óôçí ôñßôç äýíáìç) b 3 = 1 R + a + a 3 3 =1R +3a +3a 3 +3a 4 +6a 5 +4a 6 +3a 7 +3a 8 + a 9 = 1 R + a + a 3 + a 4 + a 7 + a 8 + a 9 =1 R + a + a 3 + a 4 + a + a 3 + a 4 = 1 R +(a + a 3 + a 4 )=1 R. ÅðåéäÞ b 3 = b b =1 R, Ý ïõìå b R ìå b 1 = b. Åäþ áðïðåñáôïýôáé ç áðüäåéîç! ÈÅÌÁ 7ï (i) Ãéá êüèå a + b 5 Z[ 5] (a, b Z) ïñßæïõìå ôçí åðéññéðôéêþ áðåéêüíéóç f : Z[ 5] Z 7, f(a + b 5) := [a +4b] 7. 1

2 Ãéá ïéïõóäþðïôå a, b, c, d Z Ý ïõìå êáé f((a + b 5) + (c + d 5)) = f((a + c)+(b + d) 5) = [(a + c)+4(b + d)] 7 = [a +4b] 7 +[c +4d] 7 = f(a + b 5) + f(c + d 5) f((a + b 5)(c + d 5)) = f((ac 5bd)+(ad + bc) 5) = [(ac 5bd)+4(ad + bc)] 7 = [ac 5bd] 7 +[4(ad + bc)] 7 = [ac] 7 +([ 5] 7 [bd] 7 )+[4(ad + bc)] 7 = [ac] 7 +([16] 7 [bd] 7 )+[4(ad + bc)] 7 = [ac +16bd] 7 +[4(ad + bc)] 7 = [(ac +16bd)+4(ad + bc)] 7 = [(a +4b)(c +4d)] 7 = [a +4b] 7 [c +4d] 7 = f(a + b 5)f(c + d 5), ïðüôå ç f åßíáé Ýíáò åðéìïñöéóìüò äáêôõëßùí. Óýìöùíá ìå ôï 1ï èåþñçìá éóïìïñöéóìþí äáêôõëßùí 3.3., Z[ 5]/Ker(f) = Z 7. ÅðåéäÞ f(7) = [ ] 7 =[7] 7 =[0] 7 êáé f(4 5) = [4 + 4 ( 1)] 7 =[0] 7, Ý ïõìå 7 Ker(f), 4 5 Ker(f) 7, 4 5 Ker(f). Êáé áíôéóôñüöùò ãéá ôõ üí a + b 5 Ker(f) (a, b Z)Ý ïõìå7 a +4b, ïðüôå a + b µ a +4b 5=7 4 5 b, 7 áð' üðïõ Ýðåôáé êáé ï áíôßóôñïöïò åãêëåéóìüò Ker(f) 7, 4 5. (ii) Áò õðïèýóïõìå üôé õðüñ åé êüðïéïò éóïìïñöéóìüò äáêôõëßùí ϑ : Z[X] Q[X]. ÅðåéäÞ Ý ïõìå ðñïöáíþò ϑ(1) Q[X] 1 ϑ(1) Q[X] êáé åðåéäþ ç áðåéêüíéóç f åßíáé åðéññéðôéêþ, õðüñ åé êüðïéï f(x) Z[X], ôýôïéï þóôå íá éó ýåé ç éóüôçôá ϑ(f(x)) = 1 ϑ(1). ÅðïìÝíùò, ϑ(f(x)) = ϑ(f(x)+f(x)) = ϑ(f(x)) + ϑ(f(x)) = 1 ϑ(1) + 1 ϑ(1) = ϑ(1). ÅðåéäÞ ç áðåéêüíéóç åßíáé åíñéðôéêþ, Ý ïõìå f(x) =1. ÅÜí f(x) = mx a i X i Z[X], ôüôå a 0 =1 a 0 = 1 QrZ. ôïðï! ÈÅÌÁ 8ï (á) (i) (ii) ÅÜí n Zr{0} êáé I n = hf(x)i, ãéá êüðïéï f(x) Z[X], ôüôå õðüñ ïõí ðïëõþíõìá g(x),h(x) Z[X], ôýôïéá þóôå íá éó ýïõí ïé éóüôçôåò n = f(x)g(x) êáé X = f(x)h(x). ÅðåéäÞ ï Z êáé, êáô' åðýêôáóéí êáé ï Z[X], åßíáé áêåñáßá ðåñéï Þ, Ý ïõìå 0=deg(f(X)g(X)) = deg(f(x)) + deg(g(x)) deg(f(x)) = deg(g(x)) = 0 (âë. ðñüôáóç 1.3.9), ïðüôå f(x) =a Zr{0}. ÅðéðñïóèÝôùò, 1=deg(f(X)h(X)) = deg(f(x)) + deg(h(x)) = deg(h(x)),

3 ïðüôå b, c Z : h(x) =bx + c. ÊáôÜ óõíýðåéáí, X = f(x)h(x) X = a(bx + c) ab =1,c=0 a Z = {±1}. Áðü ôçí Üëëç ìåñéü, f(x) I n := hn, Xi, ïðüôå θ 1 (X),θ (X) Z[X] : ñá {±1} 3a = f(x) =nd n, d {±1}. f(x) =nθ 1 (X)+Xθ (X) θ (X) =0 Z[X],θ 1 (X) =d Z. (ii) (i) ÅÜí n =0, ôüôå I n = hxi. ÅÜí n {±1}, ôüôå I n = Z[X]. ñá óå áìöüôåñåò ôéò ðåñéðôþóåéò ôï I n åßíáé êýñéï éäåþäåò ôïý Z[X]. (â) óôù n N, n. Èåùñïýìå ôïõò åðéìïñöéóìïýò äáêôõëßùí Z[X] 3 mx a ix i β 7 a 0 Z, Z 3 k γ 7 [k] n Z n. Ç óýíèåóþ ôïõò γ β : Z[X] Z n åßíáé Ýíáò åðéìïñöéóìüò äáêôõëßùí Ý ùí ùò ðõñþíá ôïõ ôï éäåþäåò ( m ) X Ker(γ β) = a i X i Z[X] [a 0] n =[0] n ( m ) X = a i X i Z[X] n a 0 ( Ã m! ) X = nk + X a ix i 1 Z[X] k Z = hn, Xi =: I n. i=1 Óýìöùíá ìå ôï 1ï èåþñçìá éóïìïñöéóìþí äáêôõëßùí 3.3., Z[X]/I n = Zn. (i) (ii) ÅÜí ôo I n åßíáé ìåãéóôéêü éäåþäåò ôïý Z [X], ôüôå ôï I n åßíáé ðñþôï éäåþäåò ôïý Z [X], äéüôé ï Z [X] åßíáé ìåôáèåôéêüò äáêôýëéïò ìå ìïíáäéáßï óôïé åßï (âë. èåþñçìá.5.). (ii) (iii) ÅÜí ôï I n åßíáé ðñþôï éäåþäåò ôïý Z [X], ôüôå ï ðçëéêïäáêôýëéïò Z[X]/I n êáé, êáô' åðýêôáóéí, êáé ï Z n åßíáé áêåñáßá ðåñéï Þ (âë. èåþñçìá.6.4 êáé ðüñéóìá (i)). ñá ï n åßíáé ðñþôïò áñéèìüò åðß ôç âüóåé ôïý ðïñßóìáôïò (iii) (i) ÅÜí ï n åßíáé ðñþôïò áñéèìüò, ôüôå ï Z n êáé, êáô' åðýêôáóéí, êáé ï ðçëéêïäáêôýëéïò Z[X]/I n åßíáé óþìá (âë. ðüñéóìá êáé ðüñéóìá (iii)). ñá ôï éäåþäåò I n åßíáé ìåãéóôéêü éäåþäåò ôïý Z [X] åðß ôç âüóåé ôïý (ii) ôïý ðïñßóìáôïò.6.5. ÈÅÌÁ 9ï (i) Áò õðïèýóïõìå üôé ôï a + b 6, a,b Z, åßíáé Ýíáò ãíþóéïò äéáéñýôçò ôïý (êáé áíôéóôïß ùò, ôïý 5) åíôüò ôþò Z[ 6]. Áðü ôá (iii), (vi) êáé (vii) ôþò ðñïôüóåùò Ýðåôáé üôé N a + b 6 N() = 4 N a + b 6 6= ±1 N a + b 6 = N a + b 6 =, 6= N() = 4 êáé áíôéóôïß ùò, N a + b 6 N(5) = 5 N a + b 6 6= ±1 N a + b 6 6= N(5) = 5 = N a + b 6 =5, ïðüôå a +6b =(êáé áíôéóôïß ùò, a +6b =5). ÁõôÝò ïé äéïöáíôéêýò åîéóþóåéò äåí åðéäý ïíôáé áêýñáéåò ëýóåéò. ñá ôï (êáé áíôéóôïß ùò, ôï 5) äåí äéáèýôåé ãíþóéïõò äéáéñýôåò, ïðüôå åßíáé áíüãùãï 3

4 óôïé åßï (âë. ôï (viii) ôþò ðñïôüóåùò 5.3.4). Êáô' áíáëïãßáí, åüí ôï a+b 6, a,b Z, åßíáé Ýíáò ãíþóéïò äéáéñýôçò ôïý 6, ôüôå N a + b 6 N( 6) = 10 N a + b 6 6= ±1 N a + b 6 6= N( 6) = 10 = N a + b 6 {, 5}, Þôïé êüôé ðïõ (üðùò Ý ïõìå Þäç áðïäåßîåé) åßíáé áäýíáôï. ñá êáé ôï 6 åßíáé áíüãùãï óôïé åßï ôþò Z[ 6]. Áðü ôçí Üëëç ìåñéü, Ý ïõìå 6=(i 6)( i 6) áëëü - ±i 6, ïðüôå ôï äåí åßíáé ðñþôï. Êáô' áíáëïãßáí, ôï 5 åßíáé äéáéñýôçò ôïý ( + 6)( 6) áëëü 5 - ± 6 êáé ôï 6 åßíáé äéáéñýôçò ôïý 5 áëëü 6 - êáé 6-5. ñá ôá, 5, 6 åßíáé ðñþôá óôïé åßá ôþò Z[ 6]. (ii) óôù d ÌÊÄ Z[ 6] (5, + 6). Ôüôå (ëáìâáíïìýíïõ õð' üøéí ôïý (i)) Ý ïõìå d 5 N (d) N(5) = 5 d + 6 N (d) N( + 6) = 10 N (d) ìêä(10, 5) = 5 = N (d) =1. ¼ìùò ôïýôï óçìáßíåé üôé d {±1} 1 ÌÊÄ Z[ 6] (5, + 6). Áðü ôçí Üëëç ìåñéü, 5, + 6 $Z[ 6] 1 / 5, + 6. (iii) Áò õðïèýóïõìå üôé ÌÊÄ Z[ 6] (10, 4+ 6) 6= êáé üôé d ÌÊÄ Z[ 6] (10, 4+ 6). Ôüôå d 10 N (d) N(10) = 100 d 4+ 6 N (d) N(4 + 6) = 40 ) = N (d) ìêä(100, 40) = 0. ÅðåéäÞ 10 êáé 4+ 6 Ý ïõìå d (âë. 5..9). Êáô' áíáëïãßáí, åðåéäþ êáé Ý ïõìå + 6 d. ÅðïìÝíùò, N () = 4 N (d) N( + 6) = 10 N (d) = N (d) =0. N (d) 1, N (d) 0 ÅÜí d = x + y 6, x,y Z, ôüôå x +6y =0. Ðñïöáíþò, x 4 êáé y 1 Áðü ôïí ðßíáêá üëùí ôùí äõíáôþí ôéìþí ëáìâüíïõìå x y x +6y x y x +6y ïðüôå ç x +6y =0äåí äéáèýôåé êáíýíá æåýãïò (x, y) Z ùò ëýóç ôçò. ôïðï! (iv) Ôï óôïé åßï 10 ìðïñåß íá ãñáöåß ùò åîþò üðïõ 10 = 5=(+ 6)( 6), 6 ± 6, 5 6 ± 6, óõí. óõí. üðùò äéáðéóôþíåôáé Üìåóá (ìå áðëýò ðñüîåéò). 4

5 ÈÅÌÁ 10ï (á) Äßäïíôáé ôá åîþò éäåþäç ôïý äáêôõëßïõ Q [X 1, X ]: (i) X 1, (iv) X 1 1, (ii) hx 1, X 3i, (v) hx 1 + X i, (iii) X 1 +1, (vi) hx 1X i. Èá åîåôüóïõìå ðïéá åî áõôþí åßíáé ðñþôá êáé ðïéá ìåãéóôéêü. ÅðåéäÞ ï äáêôýëéïò Q [X 1, X ] åßíáé ìåôáèåôéêüò, Ýíá éäåþäåò p $Q[X 1, X ] åßíáé ðñþôï åüí êáé ìüíïí åüí éó ýåé ç óõíåðáãùãþ [f(x 1, X )g(x 1, X ) p = åßôå f(x 1, X ) p åßôå g(x 1, X ) p], ãéá ïéáäþðïôå f(x 1, X ),g(x 1, X ) Q [X 1, X ] (âë. ðñüôáóç.5.). ÅðéðñïóèÝôùò, åðåéäþ ï Q [X 1, X ] åßíáé äáêôýëéïò ìå ìïíáäéáßï óôïé åßï, êüèå ìåãéóôéêü éäåþäåò ôïõ åßíáé êáô' áíüãêçí ðñþôï (âë. èåþñçìá.5.). (i) ÅðåéäÞ X 1 = X 1 X 1 X1 áëëü X1 / X1, ôï X 1 äåí åßíáé ïýôå ðñþôï ïýôå ìåãéóôéêü éäåþäåò. (ii) ÅÜí ïñßóïõìå ôïí ïìïìïñöéóìü äáêôõëßùí ϑ : Q [X 1, X ] Q, f(x 1, X ) 7 f(, 3), ðáñáôçñïýìå üôé ï ϑ åßíáé åðéìïñöéóìüò. (Ãéá êüèå λ Q, õðüñ åé Ýíá ðïëõþíõìï f λ (X 1, X ):=(X 1 ) + (X 3) + λ, ãéá ôï ïðïßï éó ýåé ϑ (f λ (X 1, X )) = λ). μñçóéìïðïéþíôáò ôï 1ï èåþñçìá éóïìïñöéóìþí äáêôõëßùí 3.3. ó çìáôßæïõìå Ýíáí éóïìïñöéóìü Q [X 1, X ] / Ker(ϑ) = Q. Ïåãêëåéóìüò hx 1, X 3i Ker(ϑ) ={ f(x 1, X ) Q [X 1, X ] f(, 3) = 0} åßíáé ðñïöáíþò. Ï áíôßóôñïöïò åãêëåéóìüò '' Ýðåôáé áðü ôï üôé êüèå f(x 1, X )= X i,j a ij X i 1X j Q [X 1, X ] ìå f(, 3) = 0 ìðïñåß íá ãñáöåß ùò f(x 1, X )= X i,j a ij ( + X 1 ) i (3 + X 3) j = X i,j " ix #" a i jx ij k k (X 1 ) i k k=0 l=0 # j l 3 l (X 3) j l Þôïé ùò áèñïßóìáôá ñçôþí áñéèìþí ðïëëáðëáóéáæïìýíùí ìå äõíüìåéò ôùí X 1 êáé X 3. EðåéäÞ ôï Q åßíáé óþìá, ï ðçëéêïäáêôýëéïò Q [X 1, X ] / hx 1, X 3i åßíáé ùóáýôùò óþìá êáé ôï éäåþäåò hx 1, X 3i ìåãéóôéêü (êáé, êáô' åðýêôáóéí, ðñþôï), âë. ðüñéóìá êáé ðüñéóìá (iii). (iii) ÅðåéäÞ ôï X 1 +1åßíáé áíüãùãï ðïëõþíõìï óôïí Q [X 1], ôï X 1 +1 åßíáé ðñþôï éäåþäåò (ôüóïí ôïý Q [X 1 ] üóïí êáé ôïý 1 Q [X 1, X ]), áëëü äåí åßíáé ìåãéóôéêü éäåþäåò, êáèüôé, ð.., X 1 +1 $ X 1 +1, X 3 $Q[X 1, X ]. (iv) ÅðåéäÞ X 1 1=(X 1 1)(X 1 +1), ôï X 1 1 äåí åßíáé áíüãùãï, ïðüôå ôï X 1 1 äåí åßíáé ïýôå ðñþôï ïýôå ìåãéóôéêü éäåþäåò. 1 Ôüóïí ï Q [X 1 ] üóïí êáé ï Q [X 1, X ] åßíáé Ð.Ì.Ð., ïðüôå êüèå áíüãùãï óôïé åßï ôïõò åßíáé êáé ðñþôï óôïé åßï (ôï ïðïßï ðáñüãåé Ýíá ðñþôï éäåþäåò, âë. èåþñçìá êáé ðñüôáóç (i)). 5

6 (v) Ôï X 1 + X åßíáé áíüãùãï ðïëõþíõìï êáé, ùò åê ôïýôïõ, ðñþôï óôïé åßï ôïý Q [X 1, X ], ïðüôå ôï hx 1 + X i åßíáé ðñþôï éäåþäåò. Ùóôüóï, ôï hx 1 + X i äåí åßíáé ìåãéóôéêü, äéüôé ð.. hx 1 + X i $ hx 1, X i $Q[X 1, X ]. (vi) ÅðåéäÞ X 1 X hx 1 X i áëëü X 1 / hx 1 X i êáé X / hx 1 X i, ôï hx 1 X i äåí åßíáé ïýôå ðñþôï ïýôå ìåãéóôéêü éäåþäåò. (â) Äßäïíôáé ôá åîþò óôïé åßá ôïý äáêôõëßïõ Z [X 1, X, X 3]: Èá åîåôüóïõìå ðïéá åî áõôþí åßíáé áíüãùãá. (i) ÅðåéäÞ (i) f(x 1, X, X 3 ):=X 1 + X + X 3, (ii) g(x 1, X, X 3):=X 1X + X X 3 + X 3X 1. f(x 1, X, X 3)=(X 1 + X + X 3), ìå ôï X 1 + X + X 3 ìç óôáèåñü ðïëõþíõìï, ôï f(x 1, X, X 3 ) äåí åßíáé áíüãùãï. (ii) Áò õðïèýóïõìå üôé ôï g(x 1, X, X 3) ãñüöåôáé ùò ãéíüìåíï äýï ìç óôáèåñþí ðïëõùíýìùí h 1 (X 1, X, X 3 ) êáé h (X 1, X, X 3 ) Z [X 1, X, X 3 ]. ÅðåéäÞ ôï g(x 1, X, X 3 ) äåí ðåñéý åé êáìßá ìåôáâëçôþ õøùìýíçóåêüðïéáäýíáìç êáé ï óôáèåñüò ôïõ üñïò åßíáé =0, ôá h 1(X 1, X, X 3) êáé h (X 1, X, X 3) äåí ìðïñïýí íá ðåñéý ïõí ìç ìçäåíéêïýò óôáèåñïýò üñïõò, ïýôå üñïõò ðïõ ðåñéëáìâüíïõí ðåñéóóüôåñåò ôþò ìßáò ìåôáâëçôþò, ïýôå üñïõò ðïõ ðåñéëáìâüíïõí ìßá ìåôáâëçôþ õøùìýíç óå êüðïéá äýíáìç. ÊáôÜ óõíýðåéáí, ôá h 1 (X 1, X, X 3 ) êáé h (X 1, X, X 3 ) åßíáé ôþò ìïñöþò h 1(X 1, X, X 3) = [a 1] X 1 +[a ] X +[a 3] X 3, h (X 1, X, X 3 ) = [b 1 ] X 1 +[b ] X +[b 3 ] X 3, ãéá êüðïéïõò a 1,a,a 3,b 1,b,b 3 {0, 1}. Áðü ôçí éóüôçôá Ýðåôáé üôé X 1 X + X X 3 + X 3 X 1 = h 1 (X 1, X, X 3 )h (X 1, X, X 3 ) [a 1 ] [b 1 ] =[a ] [b ] =[a 3 ] [b 3 ] =[0], [a 1 ] [b ] +[a ] [b 1 ] =[1], [a 1 ] [b 3 ] +[a 3 ] [b 1 ] =[1], [a ] [b 3] +[a 3] [b ] =[1]. ÂÜóåé ôùí ðñþôùí óõíèçêþí, åßôå a j =0åßôå b j =0, ãéá êüèå j {1,, 3}. Áò õðïèýóïõìå üôé a 1 =0. Ôüôå a = a 3 = b 1 = 1, ïðüôå b = b 3 = 0, ðñüãìá Üôïðï ëüãù ôþò ôåëåõôáßáò éóüôçôáò. Ðáñïìïßùò êáôáëþãïõìå óå Üôïðï õðïèýôïíôáò üôé a =0Þa 3 =0. Êáô' áíáëïãßáí, êáôáëþãïõìå óå Üôïðï õðïèýôïíôáò üôé Ýíáò åê ôùí b 1,b,b 3 åßíáé =0. ñá ôï g(x 1, X, X 3 ) åßíáé áíüãùãï ðïëõþíõìï