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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