Final Exam - Solutions

# Final Exam - Solutions - 1(10 pts Solve the following...

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Unformatted text preview: 1. (10 pts) Solve the following congruences. If there are multiple solutions, be sure to ﬁnd all of them. If there is no solution, say so, and be sure to justify your answer. (a) (5 pts) 8x E 11 (mod 30) . 8905!! (owl 3W¢=\$ Bﬂ/JK'N 5‘9 gx’ﬂ‘iﬂk 79wa [6% 1:? gym/<4; Par m keZ, gut Zl Elm 'Wk ml Zf/b 72) 17/13/19 I3 4a 97/:(7624, AM“le @Xflddm‘fm; (3/30)=Z/ m1 2%l/lf0 9/ WW 2,// / WM {5 M éﬂ/MHM, (b) (5 pts) 89: E 11 (mod 31) {5;5l):// 55’ 74%” "Wt £6 4 Mlztw ia/uffm Mala/ﬂ 3/; 44M 5 {(‘K" (Mal 31% 5a we mm“ am We gwgmg 93+? 7' t 4/ 96 I z I :2; 2: 8-?-/= K-(3I—3-3)-/=8"f—3// 54 ( (Mal 50 3459‘ 6494 94 a M"; Ms“ W2 (3 (M 3/) x 2 [3 (Mod 3/) 3 8 X 2. (10 pts) Let a 2 3X2 -|- X -|— 2 E Z7[X]. Compute the inverse of [a] in . H X+5 = [J' «(9(8) 2: 4 ~ (I, — 4 may - (3X) ~ : 6v [0(3X)+4 (WWW) : mime!) 7(a) NW, MW 5/ 2":9‘ +0 3% I = 2.2” = 2% : [dﬂamu 7; (320] 50 MWMEI (mm 5” E4}':?:[wz+w f [WV/w Z7[X]/(p) Where p = X3 + 4. . {Maid} wark: X . "I; ’ 5 AguAJrEd/l I P: + ~3X2+1+éﬁrj€jﬁl7ﬁl 0/‘5 MI 1 j (WW 2 “5:43;, ’ , 2x1+3x+1 Tiff {f 4 WW, {0 W We’re Jam, 07C 4 «42/0 (’51.) 3X WW WW2; {/ M I if +2 2: 4 ~ (mm) 2“ 3. (10 pts) Give an example of a ﬁeld of order 125. Be sure to explain Why your example is a ﬁeld. 125 = 53) IF f6 25 {5 6M {MAMA/a fo/[V/wmm/ 517:) 17/136 \$64 ZSDU/(lﬂ) Wf// be a an“ crazy 5, {a we Jud meal +0 7%} m [malady/x ﬂayﬂmk/ 07p a/ we 5 F Z ’4 ,W/ 3 Ol/MM/aa/ l3 WJMIZQ /4’|/C/‘ a gall {in M Ml}; 47"; if [45: M AM? In 7% f=X3/)(3+ll)<5+; Xafé/YBH/ almgf wka a- fowl/3%)”! war/<5? iw’l (1)7'3 M34 (WMH) fO/BFI if M awed MﬁV/ef. [(Lll’ﬁ‘ (HM/‘6 W6 4 0f W jlﬂfﬁ!’ Mﬂ/zfc 4ﬂ/M/l4'é I ., .L _ g = 1 #510 /V£‘L{ c,’ (2:15,! 54.56 ﬁf' fZE/"g [mg/5,4,) 4. (10 pts) Let R be a commutative ring with identity. Prove that R is an integral domain if and only if the ideal (O) is prime. (\$79) Asjme R I} an {41> Ml Jami/2, WTE (Wi/K WI 7%» any d/ééA/ otéé{0) gage/d) pr [yew]. Sim (ANN {fa/Mo, {may guy/m 496(0). Thom 4W4 M ﬂea A I; am Mfg/SJ Mm, M W H, rm 6% W) W W). Ham (0) f5 4 fr-Im {Jar/1 Cg) Agar/1e (a) 13/011146: A354 4,”? [)3 dé;ﬂ ﬁlm grifZier dip of IP H, m rsr’/=r'ﬂ:(9 ram/MA, ﬂ wiggm 517% (0} ’3 ﬁN/Wﬂ/ (Wat/Q, 40 439A, Sir/We aérﬂ Hwi aé6(ﬂ)/ 5y 7/1246 f5 Mm/ after 46 W) W éé/ﬂa 5;) 8/fo 43/ W ézﬂ Thu; A is M f/H'zym/ dip/mam, w J,’ ~-—-.«__.._.______ A / RPM/1L6 PM}?! _ ﬁdﬁ be it xiii/7% W: m» M, min )t if K {Wt/affve oMﬂ/Wgyz'liﬂ/l/ A/b/ Ker/f]: (ﬂ) fr £7 #14 W Ian/M74th memo/u 3/9) E R, / M_H / Nov TAM/W4 A 5 r V A ‘ [0) r3 far/"MQ/ 50 /R I5 4/: If/Zp/g&,144m HQ” 8 0V1 Mf 06/ Man}; 64/ (i) L? We 5. (a) (6 pts) Let R be a ring and let I and J be ideals in R. Let I+J={i+j|i€landjEJ}. Show that I + J is an ideal in R. W6 Muff 54M MW“ I+J (I) I3 Mag V/ 13’ Jazz WM ;% mad/34, ml (5/ xéjarés [makgﬂ /. OGI AW, 03*? 59 @‘ﬂMéIW; 7’” I+7 [5 Aim/77, 5; 71/; 73V 5M6 baa/«56 L,“(:zél Mdijlél 60 I +47 ’3 6195691 W {aéfmgﬂ'm . ML Aélhjl reR' TA“ 4:5ij 7E” {W6 (:5 Ué' ' go ar:(;+ji)r>ir+jréf+g [yea/am éﬁéf '5] W] M 2 db?) : F[+—{j é I Hm LL], ajagarbs fmdud'; WPWCO/‘O L’J T5 4/; (b) (7 pts) Let a, b E Z with a and b not both 0, and let d be the gcd of a and b. Show that ‘ (a) + (b) = ((1)- (Note: Here refers to the principal ideal in Z generated by the integer n, and (a) —|— (b) refers to the sum of the ideals (a) and (b) as deﬁned on the previous page.) 517146 («FEW/Ill; mJ (5)3fé5/4622 (0W): {mm 9552}, We will ﬁlm W (at/+0} 6(4) Mel (4/44) 30}! La 766(a)+(la), 71W. xsawés 79/ {me CgeZ, ﬁlm #4 My!) Al/df‘ Mal J/AS/ 5a &[/é{l‘7"é§ 50 le xé/J), Haw («2+{UCO}, / ﬂaw MW 0562 51% 74m" pl: df‘fég {by Therm L3” Jag/Mb), A/aw /@7L Ma) TM w’dk 79F Moe kéz. 5/7144 6024’ {MW} [5 4/1 Ida-é Akéé/Wl/ 60 We (42%) 3 (A). Wire (4) Mia/#4), (c) (7 pts) Let F be a ﬁeld, and let p and q be relatively prime polyno— mials in F[X Show that (P) 0 (q) = (W)- (Note: Here (f) refers to the principal ideal in F[X] generated by the polynomial f We wil/ flaw 06L) 6 owl n (6)1 L67‘ 6(6 Um are/P 74M ﬁﬂMg /‘é and, 4:15 70% 5M6- 56F[/l], NW6 [0/49 M fat/Z5] Alli 5’7Mﬂ [/7/ZFI/ we Muﬁ hit/e 3:45 4:5); 7’3» We teFZ/YJ/ 5y qszg=éff; gay; m gram/4,95). b): w “(/2 ), mm a: 7: mm téFZ/Yj/ 50 FM and Z6/4, Hwy we! ole/ﬂ ﬂ a6 @Mﬁ. Nu; (fMéﬁJ/fi}, Tim/Pom [W/Mﬂﬁm), 6. (20 pts) Recall that M2(Z) denotes the ring of 2 X 2 matrices with integer coefﬁcients. Let a b R- {(0 a) (a) (5 pts) Show that R is a commutative subring of M2(Z). Low"? “#9 4wa (geek/Q, i0 x4 [5 Mam/7 /% f)’(§ (4f fj/eﬂ/ {a A f; 05,460! arr/er iuémﬁ; m a 24 WM 60 r M «a» a,bEZ}. 4& A b j 44 «J‘féc 0 4%? 4): U M 3 ml ﬂ 2? if (’Mmhﬁi/e. (32% 2H? 42:") ﬁt / (b) (5 pts) Deﬁne a function f : R —> Z by f((3 = a. Show that f is a surjective homomorphism. m M 32/412” 2:3): at“ Mt :3) W i)(g£/):f/r “£536 : : 20/; W5 2’) 6:9 f f’WeFl/ef MMff/M WW [hf/mm} cm) W! I? ( honMorfhlk/n, [f 016%! TM 3406/5 M Ja/fisza/ 54 is 5 jaw/c, (c) (3 pts) What is the kernel of f? Km)=ﬁ3 fag/z/ m? EMZ= 5/5 f/éﬂ/ M} (d) (3 pts) Prove that R/Ker(f) 9—“ Z. 517M? 3): I3 4 fuf/éofiva N7 AMﬁMﬁ/f/ﬁfs’é; (Ab 67L ' MW 1' A/ I l L; m I/‘d/M/ R/Kw/H :7 (e) (4 pts) IS Ker( a prime ideal? Is it a maximal ideal? (Justify your answers.) 6/71“, A {9 4 MM/Vlbd’aﬁL/Ve My wifé Mia??? [We (é Well )/ MM )3 We #5” RAM} is a» 1‘4, q/AW, Mil :24 mm/ war/f) f; a {5/1 ﬂea ) 3‘: Z f! M elm/31 KIM ﬂy a £35]; if?” d PPM/16 121646” 510‘ [5 gal ...
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Final Exam - Solutions - 1(10 pts Solve the following...

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