抽象代数 孟道骥版 习题解答 第五章

ႮႿη ൞ડ๝෿đପહႵvi ∈ R(n) , 1 i nđ൐֤η (vi ) = ui ē ഡ(v1 , v2 , · · · , vn ) = (u1 , u2 , · · · , un )B, B ∈ Mn (R)đ ପહğ(u1 , u2 , · · · , un ) = (η (v1 ), · · · , η (un ))B = (u1 , u2 , · · · , un )AB đ ࠧႵAB = In đdet A · det B = 1 = det B · det Ađ‫ܣ‬det A൞Rᇏॖ୉ ჭđՖ‫ط‬η ൞R(n) ֥ሱ๝‫ܒ‬. 2) ҂၂‫ק‬ē η ൞‫ڎ‬൞R(n) ֥ሱ๝‫ܒ‬đ౼थႿRēೂ‫ݔ‬R൞ุđ ᄵη сಖ൞ሱ๝‫ܒ‬ēೂ‫ݔ‬R = Z; η : η (m) = 2m, m ∈ Z(n) đᄵႮη ҂ ൞ડഝᆩğη ҂൞ሱ๝‫ܒ‬ē 2. ഡRູࢌߐહߌēM đN ൞R-ଆēၛHom(M, N )іൕՖM ֞N ֥๝෿ ࠢ‫ކ‬ēؓη đξ ∈ Hom(M, N )đႮ(ξ + η )(x) = ξ (x) + η (x)‫ק‬ၬξ + η ē ؓa ∈ RđႮ(aη )(x) = η (ax)‫ק‬ၬaη ē൫ᆣૼğ 1) ξ + η đaη ∈ Hom(M, N )Ġ 2) ؓഈඍࡆ‫ࠣم‬RაHom(M, N )֥Ӱ‫م‬đHom(M, N )္൞R-ଆĠ 3) ೏M აN ‫ ٳ‬љ ൞ ᇇ ູmđn֥ ሱ ႮR-ଆ đ ᄵHom(M, N )൞ ᇇmn֥R-ଆē ᆣ ૼ 1) ؓ∀x, y ∈ R-ଆM đa ∈ R, (ξ + η )(ax) = ξ (ax) + η (ax) = a(ξ (x)+η (x)) = a((ξ +η )(x)); (ξ +η )(x+y ) = ξ (x)+ξ (y )+η (x)+η (y ) = 86
89 ഝ đ ‫ܣ‬λ · μ = 0đ ॖ ࡮EndZ (Qp /Z)ᇏ ໭ ਬ ၹ ሰ đ ౏ Ⴛ ൞ હ ߌ đ ‫ܣ‬EndZ (Qp /Z)൞ᆜߌ. mk 4) ّഡQp /Z൞ሱႮZ-ଆđ౏ഡ{ l + Z|mk ∈ Z, lk ∈ N, 1 k pk 1 n} ൞Qp /Z֥၂ቆࠎ. ॖ਷d = max{plk |1 k n}đ֌ d+1 + Z ∈ p mk + Z|1 k n đ઱‫؛‬Ć‫ܣ‬Qp /Z҂൞ሱႮZ-ଆ. p lk 6. ഡM ൞R-ଆđf ∈ EndR M ē 1) ᆣૼ೏f ൞ડ๝෿đᄵf ҂൞ߌEndR M ֥Ⴗਬၹሰē 2) ൫ई২ඪૼ1)֥୉ଁี҂Ӯ৫ē ᆣ ૼ 1) ّഡg, f = 0, g, f ∈ EndR M ,Ⴕg · f = 0,ࠧؓ಩ၩx ∈ M, (g · f )(x) = g (f (x)) = 0. ၹູf ൞ડ๝෿đ෮ၛf (x)ᄝؓx಩౼ൈ ॖၛ౼ђM ᇏ಩၂ჭ෍. ‫ؓܣ‬಩ၩy ∈ M, g (y ) = 0.Ֆ‫ط‬g = 0.აࡌഡ ઱‫؛‬. ෮ၛf ҂൞ߌEndR M ֥Ⴗਬၹሰ. 2) Z(1) ൞Z-ଆ, ∀n ∈ Z(1) , f (n) = 2n. ೏g = 0, g · f = 0,ᄵ∀n ∈ Z(1) , (g · f )(n) = g (f (n)) = g (2n) = 2g (n) = 0.‫ܣ‬g (n) = i, ∀n ∈ Z(1) .Ֆ ‫ط‬g = 0.઱‫؛‬.෮ၛf ҂൞ߌEndR M ֥Ⴗਬၹሰđ‫ط‬f ҂൞ડ๝෿. 7. ഡM ൞R-ଆ ēx1 , x2 , · · · , xn ∈ M ӫ ቔ ཌ ྟ ໭ ܱ ֥ đ ೂ ‫ؓ ݔ‬Rᇏ ಩ ၩn۱҂ಆູਬ֥ჭ෍a1 , a2 , · · · , an đႵ
Chapter 5 ଆ
5.1 ሱႮଆ
1. ഡRູࢌߐહߌđη ൞R(n) ֥ሱ๝෿ē 1) ൫ᆣη ೏ູડሱ๝෿ᄵсູR(n) ֥ሱ๝‫ܒ‬ē 2) ೏η ൞R(n) ֥၂၂๝෿đ൫໙η ൞‫ູڎ‬R(n) ֥ሱ๝‫?ܒ‬ ᆣૼ 1) ౼R(n) ᇏ၂ቆࠎ{u1 , u2 , · · · wk.baidu.com un }đഡ (η (u1 ), · · · , η (un )) = (u1 , u2 , · · · , un )A, ఃᇏA ∈ Mn (R).
87 (ξ + η )(x) + (ξ + η )(y )đ‫ܣ‬ξ + η ∈ Hom(M, N ). ๝ဢಸၞဒᆣē aη ∈ Hom(M, N )đ
2) οᅶଆ֥‫ק‬ၬđಸၞဒᆣHom(M, N )൞R-ଆē 3) ഡRn×m ൞Rഈn × m֥ इ ᆔ đ ؓ Ⴟ इ ᆔ ֥ ࡆ ‫ࠣ م‬Rა इ ᆔ ֥ Ӱ ‫ ܒ م‬Ӯ ၂ ۱ ଆ đ ౏ ൞ ၂ ۱ ᇇ ູn × m֥ ሱ Ⴎ ଆ ē ഡ{u1 , u2 , · · · , un }, {v1 , v2 , · · · , vn }‫ٳ‬љ൞M, N ֥၂ቆࠎđؓ಩ၩη ∈ Hom(M, N )đ਷: (η (u1 ), · · · , η (um )) = (v1 , v2 , · · · , vn )M (η ), M (η ) ∈ Rn×m .
ቔ႘ഝğη → M (η )đಸၞဒᆣη → M (η )൞Hom(M, N )֞Rn×m ֥ଆ ๝‫ܒ‬đ‫ܣ‬Hom(M, N ) ൞ᇇmn֥R-ଆē 3. ഡRູ ࢌ ߐ ᆜ ߌ đM ൞ ᇇn֥ ሱ ႮR-ଆ đu1 , u2 , · · · , un ູ ၂ ቆ ࠎ đf1 , f2 , · · · , fn ∈ M đK =< f1 , f2 , · · · , fn >൞M ֥ ሰ ଆ đ ᆣ ૼK ൞ᇇn֥ሱႮR-ଆ֒౏ࣇ֒ det(crdf1 , crdf2 , · · · , crdfn ) = 0 ౏Վൈؓ∀x = x + K ∈ M/K Ⴕ det(crdf1 , crdf2 , · · · , crdfn ) · x = 0. ᆣૼ ೏K ൞ ᇇn֥ ሱ ႮR-ଆ đ ഡ{v1 , v2 , · · · , vn }൞K ֥ ၂ ቆ ࠎ đ i nđᄵλ൞ડ֥đ‫ܣ‬λ൞ሱ ቔK ֥ሱ๝෿λ : λ(vi ) = fi , 1 i n}൞K ֥၂ቆࠎēॖ࡮đK ൞ᇇn֥ሱႮR๝‫ܒ‬đሱಖ{fi , 1 ଆ⇐⇒ ؓ಩ၩx ∈ Rn×1 , x = 0, (f1 , f2 , · · · , fn )X = 0. ࠺A = (crdf1 , crdf2 , · · · , crdfn )ē ॖ ᆩ ğK ൞ ᇇn֥ ሱ ႮR-ଆ⇐⇒ AX = 0, ∀X = 0, X ∈ Rn×1 đ‫ط‬R൞ࢌߐᆜߌđᄵႵR֥‫ٳ‬ൔთF . Ⴎ ཌྟսඔ৘ંᆩğdet A = 0 ⇐⇒ AX = 0, ∀X = 0, X ∈ F n×1 . ପ હႵdet A = 0 ⇒ AX = 0, ∀X = 0, X ∈ Rn×1 . ‫ط‬೏∀X = 0, X ∈ Rn×1 đႵAX = 0đ಩౼X ∈ F n×1 , X = 0đႮF ൞R֥‫ٳ‬ൔთđॖ 1 −1 ਷X = (b1 c− 1 , · · · , bn cn ) đପહX · c1 · · · cn = 0đ‫ط‬X · c1 · · · cn ∈ n×1 R đ‫ܣ‬AX = A(X · c1 · · · cn ) · (c1 · · · cn )−1 = 0đՖ‫ط‬det A = 0. ሸ ഈ ॖ ᆩ ğ ֒R൞ ࢌ ߐ ᆜ ߌ ൈ đA ∈ Mn (R), det A = 0 ⇐⇒ Ֆ ‫ ط‬ğK ൞ ᇇn֥ ሱ ႮR∀X = 0, X ∈ Rn×1 , AX = 0. ଆ ֒ ౏ ࣇ ֒det(crdf1 , crdf2 , · · · , crdfn ) = 0đ Վ ൈ đ ಩ ౼x = (u1 , u2 , · · · , un )X, X ∈ Rn×1 , det A · x = (u1 , u2 , · · · , un )(A · A∗ ) · X = ((u1 , u2 , · · · , un )A)(A∗ · X ) = (f1 , f2 , · · · , fn )(A∗ · X )đᆃ৚A∗ ൞ٚ ᆔA֥ϴෛइᆔđ‫ܣ‬det A · x ∈ K đՖ‫ؓط‬಩ၩx = x + K ∈ M/K đ Ⴕdet(crdf1 , crdf2 , · · · , crdfn ) · x = 0.
88 4. ഡR൞ࢌߐહߌđM ູᇇn֥ሱႮR-ଆđf ∈ EndR M đ൫ᆣf ູ၂၂ ֥ଆ๝෿֒౏ࣇ֒f ҂൞ߌEndR M ֥ቐਬၹሰē ᆣ ૼ “=⇒” ّഡf, , g ∈ EndR M ,౏f = 0, g = 0.Ⴕf · g = 0.ࠧ ؓ∀x ∈ M, f (g (x)) = 0. ∵ f ൞ଆ๝෿ ∴ f (0) = 0. ‫ط‬g = 0 ∴ ∃x0 ∈ M ,൐g (x0 ) = 0. ౏f (g (x0 )) = 0.ᄵႵf (0) = f (g (x0 )), g (x0 ) = 0აf ൞၂၂֥઱‫؛‬. ∴ f ҂൞ߌEndR M ֥ቐਬၹሰ. ّഡf ҂൞၂၂֥đᄵf ᄝ۳‫{ࠎק‬ui }༯֥इᆔA҂ॖ Ir 0 , ਷K = 0 ∗ , ఃᇏ0 ∈ ୉. ‫∃ܣ‬P, Qॖ୉đ൐P AQ = 0 0 Rn×r , ∗ = 0, K ∈ Rn×n , ᄵP AQK = 0,ਆшӰၛP −1 , ∴ A(QK ) = 0.Ⴎ Ⴟf = 0, ∴ A = 0. ೏QK = 0,ਆшӰၛQ−1 ֤֞K = 0აK ֥ྙൔ઱ ‫؛‬. ෮ၛQK = 0,‫ܣ‬թᄝg ∈ EndR M ,൐QK ູg ᄝ{ui }༯֥इᆔ. ᄵ ෮ Ⴕf = 0, g = 0,֌f · g = 0.აf ҂൞ߌEndR M ֥ቐਬၹሰ઱‫؛‬. ၛf ൞၂၂֥ଆ๝෿. ᆣи. 5. ഡp൞ ෍ ඔ ē ਷Qp = {m/pk |m ∈ Z}ē ᄵQp ൞ ࡆ ಕQ֥ ሰ ಕ ē ౏Z ⊆ Qp ēႿ൞അಕQp /Z൞၂۱Z-ଆēቔ႘ഝf ູf (x) = p xē∀x ∈ Qp /Zē൫ᆣૼğ 1) f ∈ EndZ (Qp /Z)Ġ 2) f ҂൞၂၂႘ഝĠ 3) f ҂൞ߌEndZ (Qp /Z)֥ቐਬၹሰĠ 4) Qp /Z҂൞ሱႮZ-ଆē ᆣ ૼ 1) ႮႿؓ಩ၩa ∈ Z, x, y ∈ Qp /ZđႵ: f (x + y ) = p(x + y ) = f (x) + f (y ), f (ax) = p(ax) = af (x)đ‫ܣ‬f ∈ EndZ (Qp /Z). 1 2) Ⴎf ( + Z) = f (0 + Z)đॖ֤f ҂൞၂၂႘ഝ. p 3) ൌ࠽ഈđ໡ૌॖၛᆣૼğEndZ (Qp /Z)൞ᆜߌ. 1 ؓ಩ၩ٤ਬჭλ ∈ EndZ (Qp /Z)đ਷d = sup{l| l + Z ∈ λ(Qp /Z)}đ p ∞ 1 +Zđ ೂ‫ݔ‬d = ∞đཁಖλ൞ડഝ.೏d < +∞đႮQp /Z = k k=1 p ml 1 ᆩğ{l|ϕ( l + Z) = d + Z, p ml }൞໭౫ࠢđᄵॖ౼l1 , l2 ∈ Nđ p p m l2 1 ml 1 ౏l2 > l1 đϕ( l2 + Z) = d + Z, ϕ( l11 + Z) = d + ZđႮ(ml2 , pl2 ) = p p p p ml l2 1đ ॖ ౼n ∈ Zđ ൐n · ml2 ≡ 1 (mod p )đ Ֆ ‫ ط‬ğϕ( l11 + Z) = p m m 1 · n l l 2 1 ϕ(ml1 · pl2 −l1 · n · ( l2 + Z)) = d−(l2 −l1 ) + Z = d + Zđ઱‫؛‬Ć‫ܣ‬d = ∞đ p p p Ֆ ‫ط‬λ၂ ‫ ק‬൞ ડ ഝ.ପ હEndZ (Qp /Z)ഈ ಩ ਆ ٤ ਬ ჭλ, μ, λ · μಯ ൞ ડ “⇐=”