А.В. Михалёв - Лекции по высшей алгебре (1106006), страница 5

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ˆâ ª, G(a) —¯à®áâ ï £à㯯 ⮣¤ ¨ ⮫쪮 ⮣¤ , ª®£¤ |G| = O(a) = p.‡ ¬¥ç ­¨¥ 2. …᫨ |G| = pk , k > 1 — ª®­¥ç­ ï p-£à㯯 ¨§ pk , k > 1, í«¥¬¥­â®¢, â® G ­¥ï¢«ï¥âáï ¯à®á⮩. „¥©á⢨⥫쭮, e 6= Z(G) ⊳ G.’¥®à¥¬ ® ª« áá¨ä¨ª 樨 ª®­¥ç­ëå ¯à®áâëå £à㯯, ¢¨¤¨¬®, § ¢¥à襭 , ¥ñ ¯®«­®¥ á¢ï§­®¥ ¤®ª § ⥫ìá⢮ ᮧ¤ ¥âáï.Œë ¤®ª ¥¬ ⥮६㠮 ⮬, çâ® ¯à¨ n ≥ 5 £à㯯 An ï¥âáï ¯à®á⮩ (¢ ç áâ­®áâ¨, A5 —¯à®áâ ï £à㯯 ). §¡¨¥­¨¥ ­ ª« ááë ᮯà省­ëå í«¥¬¥­â®¢ ¢ £à㯯¥ Sn.‡ ¬¥ç ­¨¥.

—¨á«® 横«®¢ ¤«¨­ë k ¢ Sn à ¢­®(k − 1)! (¢ë¡®à ®à¡¨âë ¨§ k í«¥¬¥­â®¢;­ ¨¬¥­ì訩 í«¥¬¥­â i1 ¢ ®à¡¨â¥; ¯¥à¥áâ ­®¢ª¨ ¨§ ®áâ «ì­ëå k − 1 í«¥¬¥­â®¢ ­ 2, . . . , k ¬¥áâ å).. (a, b, c) = (b, c, a) = (c, a, b), (a, b, c) 6= (a, c, b); (1, 2, 3), (1, 3, 2).ƒà㯯 S3 ( ¡¥«¥¢ , Z(S2 ) = S2 ), ¤¢ ®¤­®í«¥¬¥­â­ëå ª« áá ᮯà省­ëå í«¥¬¥­â®¢.ƒà㯯 S3 , |S3 | = 6.(a) (a, b) (a, b, c)1 + 3 + 2 =6nkDZਬ¥àƒà㯯 S4 , |S4 | = 24.(a) (a, b)(c, d) (a, b) (a, b, c) (a, b, c, d)1 +3+ 6 + 8 +6= 2415‹¥ªæ¨ï 11http://mmresource.nm.ru/ƒà㯯 S5 , |S5 | = 120.(a) (a, b) (a, b, c) (a, b, c, d) (a, b, c, d) (a, b)(c, d) (a, b, c)(d, e)1 + 10 + 20 +30+24 +15+20= 120kkkkkk 55555 3 1523 2!4 3!5 4!2 2 23 2!2! = 20.

—¨á«® 横«®¢ (a, b, c) à ¢­® 53 2! = 20; (d, e) = (e, d) — ¢â®¬ â¨ç¥áª¨ ®á⠢訥áï í«¥¬¥­âë .‘«ãç © (a,b)(c, d): 52 32 12 = 15. —¨á«® 横«®¢ (a, b) à ¢­® 52 , ¨§ ®áâ ¢è¨åáï 3 í«¥¬¥­â®¢ ¢ë¡®à(c, d) ¤ ¥â 32 ¢®§¬®­®áâ¨, ­® (a, b)(c, d) = (c, d)(a, b), ¨â ª 302 = 15.‘«ãç © (a, b, c)(d, e):53ƒà㯯 A5.ˆ§ 横«®¢®£® à §«®¥­¨ï ¢ S5 ®áâ îâáï (ª ª ç¥â­ë¥) á«¥¤ãî騥 ¢®§¬®­®áâ¨:(a, b)(c, d), (a, b, c, d, e).  §¡¥à¥¬ í⨠á«ãç ¨:‘«ãç © (a): 1.‘«ãç © (a, b, c):(a), (a, b, c),20 (ª ª ¨ ¢ S5 ).

DZãáâì, ¤«ï ¯à®áâ®âë, (a, b, c) = (1, 2, 3). |CS5 (1, 2, 3)| =|S5 |/| OrbS5 (1, 2, 3)| = 120/20 = 6, CS5 (1, 2, 3) = {e, (1, 2, 3), (1, 3, 2), (4, 5), (1, 2, 3)(4, 5), (1, 3, 2)(4, 5)}(DZ®ïá­¥­¨¥: H = {e, (1, 2, 3)(1, 3, 2)}, K = {e, (4, 5)}; HK ⊆ CS5 (1, 2, 3); |HK| = |H||K|/|H ∩ K| =3 · 2/1 = 6, ¯®í⮬ã CS5 (1, 2, 3) = HK). Žáâ ¢«ïï «¨èì ç¥â­ë¥: CA5 (1, 2, 3) = {e, (1, 2, 3), (1, 3, 2)},|CA5 (1, 2, 3)| = 3. Žâáî¤ , OrbA5 (1, 2, 3) = |A5 |/3 = 60/3 = 20.„®ª § ⥫ìá⢮ (¢â®à®¥). | OrbA5 (1, 2, 3)| = 20. .

DZਠn ≥ 5 «î¡ë¥ ¤¢ 3−横« ¢ £à㯯¥ An ᮯà省ë.„®ª § ⥫ìá⢮. DZãáâì σ1 = (1, 2, 3), σ2 = (a, b, c) ∈ An , n ≥ 5.  ©¤¥âáï τ ∈ Sn , ¤«ï ª®â®à®©σ2 = τ (1, 2, 3)τ −1 . ) …᫨ τ ∈ An , â® ¢á¥ ¤®ª § ­®;¡) …᫨ τ ∈ Sn \ An , â® ρ = τ (4, 5) ∈ An , (4, 5) ∈ CA5 ((1, 2, 3)). ’®£¤ , ρ(1, 2, 3)ρ−1 =τ (4, 5)(1, 2, 3)(4, 5)−1 τ −1 = τ (1, 2, 3)τ −1 = σ2 . ‘«ãç © (a, b)(c, d): | OrbA5 ((a, b)(c, d))| = 15. DZãáâì β = (1, 2)(3, 4).

’®£¤ |CS5 (β)| = 120/15 = 8 ¨­¥ ¤¥«¨â ç¨á«® |An | = 60. ’ ª¨¬ ®¡à §®¬, CA5 (β) — ᮡá⢥­­ ï ¯®¤£à㯯 ¢ CS5 (β), ¢®§¬®­ë¥¥ñ ¯®à浪¨: 4, 2, 1 (¯® ⥮६¥ ‹ £à ­ ). ® | OrbA5 (β)||CA5 (β)| = 60. ’ ª ª ª | OrbA5 (β)| ≤| OrbS5 (β)| = 15, â® | OrbA5 (β)| = 15, ¨ ¯®í⮬ã OrbA5 (β) = OrbS5 (β).„®ª § ⥫ìá⢮ (¢â®à®¥). | OrbA5 ((a, b)(c, d))| = 15.. DZ®¤áâ ­®¢ª¨ ¢¨¤ (1, 2)(3, 4) ¨ (a, b)(c, d) ᮯàï¥­ë ¢ An ¯à¨ n ≥ 5.„®ª § ⥫ìá⢮. DZãáâì m = 5 (®â«¨ç­ë© ®â 1, 2, 3, 4).

’®£¤ (3, 4, m)(1, 2)(3, 4)(3, 4, m)−1 =(1, 2)(4, m). ‘«ãç © (a, b, c, d, e): | OrbA5 (a, b, c, d, e)| = 12.‡ ¬¥â¨¬, çâ® ¤«ï α = (1, 2, 3, 4, 5): | OrbS5 (1, 2, 3, 4, 5)| = 24,|CS5 (1, 2, 3, 4, 5)| = 120/24 = 5. ’ ª ª ª (α) = {e, α, α2 , α3 , α4 } ⊆ An , â® (α) ⊆ CA5 (1, 2, 3, 4, 5) ⊆CS5 (1, 2, 3, 4, 5), ¨ ¯®í⮬ã CA5 (1, 2, 3, 4, 5) = CS5 (1, 2, 3, 4, 5). ® ⮣¤ | OrbA5 (α)| = 60/5 = 12..·OrbS5 (1, 2, 3, 4, 5) = OrbA5 (1, 2, 3, 4, 5) ∪ OrbA5 (1, 2, 3, 5, 4).(A5 ).‹¥¬¬ ‹¥¬¬ ‘«¥¤á⢨¥’¥®à¥¬ ® à §¡¨¥­¨¨ ­ ª« ááë ᮯà省­ëå í«¥¬¥­â®¢ £à㯯ë(a) (a, b, c) (a, b)(c, d) (1, 2, 3, 4, 5) (1, 2, 3, 5, 4)1 + 20 +15+12+12= 60’¥®à¥¬ .A5 — ¯à®áâ ï (­¥ª®¬¬ãâ ⨢­ ï £à㯯 ), |A5 | = 60.„®ª § ⥫ìá⢮.

DZãáâì {e} 6= H ⊳ A5 .16‹¥ªæ¨ï 11http://mmresource.nm.ru/‘«ãç © 1: DZãáâì (a, b, c) ∈ H. ’®£¤ ¨ ¢á¥ ᮯà省­ë¥ á ­¨¬ 横«ë ¤«¨­ë 3 «¥ â ¢ H, ¨å20 èâãª.|H| ≥ 35,’ ª ª ª (3, 2, 4)(1, 3, 2) = (1, 2)(3, 4), â® ¢ H «¥ â ¨ ¢á¥ 15 ¯®¤áâ ­®¢®ª (a, b)(c, d), ¨â ª­® |H| — ¤¥«¨â¥«ì ç¨á« |A5 | = 60, â. ¥. |H| = 35, ¨ ¯®í⮬ã H = A5 .‘«ãç © 2: α = (a, b, c, d, e) ∈ H. ’®£¤ (a, b)(c, d)(a, b, c, d, e)(a, b) (c, d) = (b, a, d, c, e) ∈ H,(b, a, d, c, e)(a, b, c, d, e) = (b, e, d) ∈ H, ¨ ¯®í⮬ã (á«ãç © 1) H = A5 .‘«ãç © 3: (a, b)(c, d) ∈ H. ’®£¤ (a, b, c, e, d) = (d, e)(a, c)(c, d) (a, b) ∈ H, ¨ (á«ãç © 2) ¯®í⮬ãH = A5 .‘«¥¤á⢨¥ 1.A5 — ¯à®áâ ï ­¥ª®¬¬ãâ ⨢­ ï ­¥à §à¥è¨¬ ï £à㯯 , |A5 | = 60.[A5 , A5 ] = A5 2.

G = A5 , |A5 | = 60 = 30 · 2, ­® ¢ G ­¥â ¯®¤£àã¯¯ë ¨§ 30 í«¥¬¥­â®¢ (â. ¥. ¯à¨¬¥à,ª®£¤ ®¡à 饭¨¥ â¥®à¥¬ë ‹ £à ­ ­¥ ¨¬¥¥â ¬¥áâ ).„®ª § ⥫ìá⢮. …᫨ H ⊂ G, |H| = 30, â® [G : H] = 2 ¨ H ⊳ G, çâ® ¯à®â¨¢®à¥ç¨â ¯à®áâ®â¥£à㯯ë G. 1. DZãáâì n ≥ 5, {e} 6= H ⊳ Sn , H 6= Sn , ⮣¤ H = An .2. DZãáâì O(n) = {A ∈ Mn (R)|At = A−1 }, DZãáâì SO(n) = {A ∈ O(n)||A| = 1}.„®ª § âì, çâ® SO(3) — ¯à®áâ ï £à㯯 .„®ª § ⥫ìá⢮.‘«¥¤á⢨¥“¯à ­¥­¨¥“¯à ­¥­¨¥17‹¥ªæ¨ï 13http://mmresource.nm.ru/‹…Š–ˆŸ13.26­®ï¡àï 2002 £.‚ëç¨á«¥­¨¥ ¢ ä ªâ®àª®«ìæ¥ ª®«ìæ ¬­®£®ç«¥­®¢¯® £« ¢­®¬ã ¨¤¥ «ã, ¯®à®¤¥­­®¬ã ­¥¯à¨¢®¤¨¬ë¬¬­®£®ç«¥­®¬DZãáâì K — ¯®«¥, K[x] — ª®«ìæ® ¬­®£®ç«¥­®¢, p(x) — ­¥¯à¨¢®¤¨¬ë© ¬­®£®ç«¥­, deg p(x) = n,(p(x)) = K[x]p(x) = {f (x)p(x)|f (x) ∈ K[x]}.

ƒ« ¢­ë© ¨¤¥ « ª®«ìæ K[x], ¯®à®¤¥­­ë© ¬­®£®ç«¥­®¬ p(x), L = K[x]/(p(x)) — ä ªâ®àª®«ìæ®. Œë ãáâ ­®¢¨«¨, çâ® L — ¯®«¥ ¨ ®â®¡à ¥­¨¥, ¯à¨ª®â®à®¬ k → k̄ = k + (p(x)), ®áãé¥á⢫ï¥â ¢«®¥­¨¥ ¯®«ï K ¢ ¯®«¥ L.…᫨ h(x) ∈ K[x], â® h(x) = p(x)q(x)+r(x), ¯à¨ í⮬ r(x) = k0 +k1x+. . .+kn−1 xn−1 , n = deg p(x),k0 , k1 , .

. . , kn−1 ∈ K. DZ®í⮬ãh(x) + (p(x)) = r(x) + p(x)q(x) + (p(x)) = r(x) + (p(x)) == k0 + k1 x + . . . + kn−1 xn−1 + (p(x)) = k̄0 + k̄1 ξ + . . . + k̄n−1 ξ n−1 ,£¤¥ ξ = x + (p(x)) ∈ L.’ ª¨¬ ®¡à §®¬:1) r(x) — ª ­®­¨ç¥áª¨© ¯à¥¤áâ ¢¨â¥«ì ᬥ­®£® ª« áá h(x) + (p(x)) (®¤­®§­ ç­® ®¯à¥¤¥«¥­­ë©, ª ª ¨¬¥î騩 ­ ¨¬¥­ìèãî á⥯¥­ì).2) Š®«ìæ¥¢ë¥ ®¯¥à 樨 á í«¥¬¥­â ¬¨ ¯®«ï L ¬®­® ¨­â¥à¯à¥â¨à®¢ âì ª ª: ) á«®¥­¨¥ ¬­®£®ç«¥­®¢ r(x), deg r(x) ≤ n − 1; ¡) 㬭®¥­¨¥ á ¯®á«¥¤ãî騬 ¢§ï⨥¬ ®áâ ⪠®â ¤¥«¥­¨ï ­ p(x)(â. ¥. ¥á«¨ r1 (x)r2 (x) = p(x)q(x) + r3 (x), deg r1 (x) ≤ n − 1, deg r2 (x) ≤ n − 1, deg r3 (x) ≤ n − 1, â®(r1 (x) + (p(x)))(r2 (x) + (p(x))) = r3 (x) + (p(x)) ).3) ‘ç¨â ï, çâ® K ⊆ L (¯à¨ ®â®¤¥á⢫¥­¨¨ K ∋ k → k̄ = k + (p(x)) ∈ L), ¬ë ¢¨¤¨¬, çâ®{1, ξ, .

. . , ξ n−1 }, £¤¥ ξ = x + (p(x)) ∈ L, — ¡ §¨á «¨­¥©­®£® ¯à®áâà ­á⢠K L ­ ¤ ¯®«¥¬ K, ¨, â ª¨¬®¡à §®¬, dimK L = n.4) ’ ª ª ª p(ξ) = 0, â® ­¥¯à¨¢®¤¨¬ë© ¬­®£®ç«¥­ p(x) ¨¬¥¥â ¢ à áè¨à¥­­®¬ L ¯®«ï K ª®à¥­ì.. K = R — ¯®«¥ ¤¥©á⢨⥫ì­ëå ç¨á¥«, p(x) = x2 + 1 ∈ R[x] — ­¥¯à¨¢®¤¨¬ë© ¬­®£®ç«¥­. ’®£¤ L = R[x]/(x2 + 1) ∼= C (C — ¯®«¥ ª®¬¯«¥ªá­ëå ç¨á¥«).„®ª § ⥫ìá⢮ 1-¥. ’ ª ª ª ξ 2 + 1 = 0, ¤«ï ξ = x + (x2 + 1), â® ¤«ï a, b ∈ R ᮮ⢥âá⢨¥a + bξ ↔ (a, b) ↔ a + bi ï¥âáï ¨§®¬®à䨧¬®¬ ¯®«¥© L = R[x]/(x2 + 1) ¨ C.

„®ª § ⥫ìá⢮ 2-¥.  áᬮâਬ £®¬®¬®à䨧¬ ∆ : R[x] → C, ¤«ï ª ¤®£® ∆(f (x)) = f (i).‡ ¬¥â¨¬, çâ® ker ∆ = {f (x) ∈ R[x]|f (i) = 0} = R[x](x2 + 1) = (x2 + 1), ¯®áª®«ìªã ¥á«¨ ¬­®£®ç«¥­f (x) á ¤¥©á⢨⥫ì­ë¬¨ ª®íää¨æ¨¥­â ¬¨ ¨¬¥¥â ª®à¥­ì i, â® −i â ª¥ ï¥âáï ¥£® ª®à­¥¬, ¨¯®í⮬ã f (x) ¤¥«¨âáï ­ (x − i)(x + i) = x2 + 1. DZ® ⥮६¥ ® £®¬®¬®à䨧¬¥ ¤«ï ª®à­ïDZਬ¥àL = R[x]/(x2 + 1) ∼= C.“¯à ­¥­¨¥. DZãáâì p(x) ∈ R[x] — «î¡®© ­¥¯à¨¢®¤¨¬ë© ¬­®£®ç«¥­ 2-© á⥯¥­¨.

’®£¤ L = R[x]/(p(x)) ∼= C.DZਬ¥à. DZãáâì K = Zp — ¯®«¥ ¢ëç¥â®¢ ¯® ¬®¤ã«î p, |K| = p, p(x) ∈ Zp [x] — ­¥¯à¨¢®¤¨¬ë©¬­®£®ç«¥­, deg p(x) = n. ’®£¤ L = Zp [x]/(p(x)) — ª®­¥ç­®¥ ¯®«¥ ¨§ pn í«¥¬¥­â®¢.‚ ç áâ­®áâ¨, ¯ãáâì K = Z2 , p(x) = x2 + x + 1 ∈ Z2 [x] — ­¥¯à¨¢®¤¨¬ë© ¬­®£®ç«¥­, ⮣¤ L = Z2 [x]/(x2 + x + 1) — ¯®«¥ ¨å 4-å í«¥¬¥­â®¢.“¯à ­¥­¨¥. DZ®áâநâì ª®­¥ç­®¥ ¯®«¥ ¨§ 8 ¨ ¨§ 9 í«¥¬¥­â®¢.18‹¥ªæ¨ï 13http://mmresource.nm.ru/DZà®á⮥ ¯®¤¯®«¥Ž¯à¥¤¥«¥­¨¥. DZãáâì L — ¯®«¥. DZà®á⮥ ¯®¤¯®«¥ — íâ® ­ ¨¬¥­ì襥 ¯®¤¯®«¥, ᮤ¥à 饥 1(®­®áãé¥áâ¢ã¥â, ª ª ¯¥à¥á¥ç¥­¨¥ ¢á¥å ¯®¤¯®«¥© ¯®«ï L). áᬮâਬ ª ­®­¨ç¥áª¨© £®¬®¬®à䨧¬ ª®«¥æ∆ : Z → Z · 1 ⊆ L,¯à¨ ª®â®à®¬ ∆(n) = n · 1 ¤«ï n ∈ Z.‚®§¬®­ë ¤¢ á«ãç ï:‘«ãç © 1: char L = 0.

‚ í⮬ á«ãç ¥ ker ∆ = 0, ¨ ¯®í⮬ã ∆ : Z → Z · 1, n → n · 1, — ¨§®¬®à䨧¬ª®«¥æ. ’ ª¨¬ ®¡à §®¬, ­ ¨¬¥­ì襥 ¯®¤¯®«¥ ¯®«ï L, ᮤ¥à 饥 1, á®á⮨⠨§ í«¥¬¥­â®¢ (m · 1)(n ·1)−1 , n 6= 0, ¨ ¨§®¬®àä­® ¯®«î Q à 樮­ «ì­ëå ç¨á¥«.‘«ãç © 2: char L = p — ¯à®á⮥ ç¨á«®. ’®£¤ ker ∆ = Zp , ¨ ¯® ⥮६¥ ® £®¬®¬®à䨧¬ åZ·1 ∼= Z/Zp ∼= Zp ,â. ¥.

Z · 1 = {n · 1|n ∈ Z}, ®ª §ë¢ ¥âáï ­ ¨¬¥­ì訬 ¯®¤¯®«¥¬ ¢ L, ᮤ¥à 騬 1, ¨ ¨§®¬®àä­®¯®«î ¢ëç¥â®¢ Zp ¤«ï p = char L.‘«¥¤á⢨¥.1)2)¥á«¨ char L = 0, â® ¬®­® áç¨â âì, çâ® ¯®«¥ L ᮤ¥à¨â ¯®«¥ Q à 樮­ «ì­ëå ç¨á¥«;¥á«¨ char L = p, â® ¬®­® áç¨â âì, çâ® ¯®«¥ L ᮤ¥à¨â ¯®«¥ ¢ëç¥â®¢ Zp .‘â஥­¨¥ ª®­¥ç­ëå ¯®«¥©’¥®à¥¬ .…᫨ L — ª®­¥ç­®¥ ¯®«¥, p = char L, â® |L| = pn .„®ª § ⥫ìá⢮. DZ®«¥ L c®¤¥à¨â ¯®«¥ ¢ëç¥â®¢ Zp (ª ª ¯à®á⮥ ¯®¤¯®«¥), Zp⊆ L, ¯®í⮬㪮­¥ç­®¬¥à­®¥ «¨­¥©­®¥ ¯à®áâà ­á⢮ ­ ¤ ¯®«¥¬ Zp .

’ ª¨¬ ®¡à §®¬, ¥á«¨ {e1, . . . , en} —¡ §¨á «¨­¥©­®£® Zp -¯à®áâà ­á⢠L, â® ª ¤ë© í«¥¬¥­â ¯®«ï L ®¤­®§­ ç­® ¯à¥¤áâ ¢«ï¥âáï ¢ ¢¨¤¥L —λ1 e1 + . . . + λn en , λi ∈ Zp ,¨ ¯®í⮬ã |L| = pn . ‡ ¬¥ç ­¨¥. ¥ áãé¥áâ¢ã¥â ª®­¥ç­ëå ¯®«¥© ¨§ 6 í«¥¬¥­â®¢ (6 = 2 · 3 ­¥ ¨¬¥¥â ¢¨¤ pn ¢ à §«®¥­¨¨ ­ ¯à®áâë¥ ¬­®¨â¥«¨). 襩 á«¥¤ãî饩 § ¤ 祩 ï¥âáï ¤®ª § ⥫ìá⢮ â¥®à¥¬ë ® ⮬, çâ® ¤«ï «î¡®£® ç¨á« ¢¨¤ N = pn áãé¥áâ¢ã¥â (¨ ¥¤¨­á⢥­­®¥, á â®ç­®áâìî ¤® ¨§®¬®à䨧¬ ) ª®­¥ç­®¥ ¯®«¥ ¨§ N = pní«¥¬¥­â®¢.’¥®à¥¬ .Œã«ì⨯«¨ª ⨢­ ï £à㯯 L∗£à㯯®©.„®ª § ⥫ìá⢮.

DZãáâì |L| = N¤«ï ª®­¥ç­ëå ¡¥«¥¢ëå £à㯯= pn ,= L \ {0}ª®­¥ç­®£® ¯®«ï L ï¥âáï 横«¨ç¥áª®©â®£¤ m = |L∗| = N − 1. ‚ ᨫã áâàãªâãà­®© ⥮६ëL∗ = G1 × . . . × Gr , |Gi | = pki i ,Gi = (ai ) — ¯à¨¬ à­ ï 横«¨ç¥áª ï £à㯯 .…᫨ pi = pj ¤«ï i 6= j, â® t = ŽŠ(pk11 , . . . , pkr r ) < pk11 . . . pkr r = m, ¨ ⮣¤ xt = 1 ¤«ï ¢á¥å x ∈ L∗,∗|L | = m > t, â. ¥.

¬­®£®ç«¥­ ­ ¤ ¯®«¥¬ L ¨¬¥¥â à §«¨ç­ëå ª®à­¥© ¡®«ìè¥, 祬 ¥£® á⥯¥­ì, çâ®­¥¢®§¬®­®.ˆâ ª, pi 6= pj ¯à¨ i 6= j, ¨ ⮣¤ L∗ = G1 × . . . × Gr , |Gi | = pki i , — 横«¨ç¥áª ï £à㯯 (ª ª¯à®¨§¢¥¤¥­¨¥ 横«¨ç¥áª¨å £à㯯 ¢§ ¨¬­® ¯à®áâëå ¯®à浪®¢). 19‹¥ªæ¨ï 13http://mmresource.nm.ru/‘«¥¤á⢨¥. Z∗p = Zp \ {0} — 横«¨ç¥áª ï £à㯯 ¨§ p − 1 í«¥¬¥­â .‡ ¬¥ç ­¨¥. ‚ë¡à ¢ 横«¨ç¥áª¨© ®¡à §ãî騩 ¢ ¬ã«ì⨯«¨ª ⨢­®© £à㯯¥ L∗ ª®­¥ç­®£® ¯®«ïL ¨ § ¯¨á ¢ ­¥­ã«¥¢ë¥ í«¥¬¥­âë ¯®«ï L ª ª ¥£® á⥯¥­¨, ¯®«ãç ¥¬ 㤮¡­ë© ¨ íä䥪⨢­ë© ᯮᮡ㬭®¥­¨ï í«¥¬¥­â®¢ ª®­¥ç­®£® ¯®«ï.’à㤭 ï § ¤ ç . (⥮६ ‚¥¤¤¥à¡ à­ ): ª®­¥ç­®¥ ⥫® ª®¬¬ãâ ⨢­®, â. ¥. ï¥âáï ª®­¥ç­ë¬¯®«¥¬ (¨­ë¬¨ á«®¢ ¬¨, ª®­¥ç­ëå ⥫ ­¥ áãé¥áâ¢ã¥â)..

DZãáâì L — ª®­¥ç­®¥ ¯®«¥, char L = p, N = |L| = pn , ⮣¤ aN = a ¤«ï ¢á¥å a ∈ L,â. ¥. ¢á¥ í«¥¬¥­âë a ∈ L ïîâáï à §«¨ç­ë¬¨ ª®à­ï¬¨ ¬­®£®ç«¥­ xN − x ∈ Zp [x], ¨ ¯®í⮬ãxN − x = (x − a1 ) . . . (x − aN ), £¤¥ a1 , . . . , aN — ¢á¥ à §«¨ç­ë¥ í«¥¬¥­âë ¯®«ï L.„®ª § ⥫ìá⢮. ’ ª ª ª |L∗ | = N − 1, â® aN −1 = 1 ¤«ï ¢á¥å 0 6= a ∈ L, ¯®í⮬ã aN − a = 0 ¤«ï¢á¥å a ∈ L. rrrrrr. …᫨ L — ¯®«¥, char L = p, â® (a + b)p = ap + bp , (ab)p = ap bp ¤«ï ¢á¥å r ≥ 1, â. ¥.¢®§¢¥¤¥­¨¥ ¢ p-ãî á⥯¥­ì ï¥âáï £®¬®¬®à䨧¬®¬¨§ L ¢ L.P„®ª § ⥫ìá⢮. ’ ª ª ª (a + b)p = pk=0 Cpk ak bp−k ¨ ¡¨­®¬¨­ «ì­ë¥ ª®íää¨æ¨¥­âë Cpk ¯à¨r0 < k < p ¤¥«ïâáï ­ p, â® (a + b)p = ap + bp . ˆâ¥à¨àãï íâ® á®®¡à ¥­¨¥, ¯®«ãç ¥¬, çâ® (a + b)p =rrrrrap + bp ¯à¨ r ≥ 1. Ÿá­®, çâ® (ab)p = ap bp ¤«ï «î¡®£® k ∈ N.

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