absalg08s_ex1

# absalg08s_ex1 - NAME: Abstract Algebra Math 521A...

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Unformatted text preview: NAME: Abstract Algebra Math 521A N-‘lichael E. O’Sullivan First Exam Friday. Feb 1:"). "2008 DO all problems. Please Show your work! Good luck. 1. [20 pts.] 0110 of the key lEEI’EIIIIELS we used the following: Let a, b. and C be integers. If a. divides b and b dividoS (: then a divides c. Prove this result using the deﬁnition of ﬁlvides. 3. In ibis problem we work in Z m. (a) [35 pm] Find all zero divisors in 210 and find mi! possible pairs (Lb such that. a and b are nonzero but the product ab :- 0. (b) [15 pts.] Find ali invertible Gimme-miss and their izwerses. g w 5‘ x *7. Wm 1 . ‘ A f. at 113%. 3m 6%. a; a: 4:: m ‘2. ":3" , m 2 ‘ 9:3 3’ , x J, - f . X :21; 3:3 \$53 :M 3W; kaixwa. I: A; a “a _ A 5 ; f. ‘ i 3 3 5i? .. :5 “2‘3 ) L f E g 5 fmia. Va "£4 » ésv K K. % “Mr i "C; i “:3? M. m g; ~~ "75 W? .. "f x“ ﬂ 3'" f i7 4. in this problem we look at: solutions of linear“ equations in Z15. (a) [10 pts] Find all soluticms to 4:1: 2 7 in 215. (b) [10 ms] Give an exmnple of a linear equatioz‘l in 215 {hat has no solutions. Explain why it has no mint-ions. (C) [15 ptsl Give an example of a linear ecqilaéigm in 2715 that has 1120a? than one solution, and identify all of the solutions. 5‘}, R‘» r s “ ii: i wav- Eggﬁeﬁséiww i f: 193‘ a“? if? j ‘33:} WW v g i *3 “m 22:: gm... «‘7; Maw/[B ‘x m M A if“ ‘ ’3 4: g l {’2 f 1) g)“ g M’s/5.3% w :3 g u) M g: i: v E ‘1 l N? f ’ vu—x 5" W _ _ s; ” l”? 4i? a 4H a w z ‘ i S K; 3 J 5. Consider the equation (1'2 + 3,2? -:r: U in 255”. (a) [15 Prove that. if *n is prime and n. > 3 Eben there are exactly two distinct solutions. (b) {15 pts.] Show by example. that more can be more than two soEutions if n is composite. (Try some Slnail composite numbers 'n.) 6. [20 p23] Use the Euclidean algorithm to ﬁnd the inverse of 1.7 in Z49. § 3 z"- “‘2 f M K W kg: m m g M: ‘1 g, 2“ x f W, A 1w ; ¢ 2; M4 ,F “"2 w i f , j} 5‘ T?“ “" w} M. w W ,y w s f? M} “” l” ' /’“ “v.1 W? ’4‘ M w” 3‘ E M 34 63;” M 93% {3: ‘E 2‘ If“ <: x" “ w W ’5‘ R if a) ‘33 NM 3 f M é/x} 3 1 M3 gm». J’ «a row; ~ 1 = 6/. g Tér (:3. ' »W A “5 i2 g g "‘P " is a 4/ W {i 3 g 5:? if 2: J w .5 ; 5w ’" f ...
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## This note was uploaded on 02/04/2010 for the course MATH521A n/a taught by Professor Michaelo'sullivan during the Fall '09 term at San Diego State.

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absalg08s_ex1 - NAME: Abstract Algebra Math 521A...

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