Midterm 2 - Solutions

Midterm 2 - Solutions - 1. Let R be a ring. Recall that an...

Info iconThis preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 2
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 4
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 1. Let R be a ring. Recall that an element a of R is called m'lpotent if there exists a positive integer n such that a" 2 OR. (a) Explain what it would mean to say that the property of being a nilpotent element of R is “preserved by homomorphisms”. PM 4A; my WWW/Mm I?” at I? 4&0fo in R 4 JV“) [5 MVNEIUL {/1 g (b) Prove the statement you made in part (a). (In other words, prove that being a nilpotent element of R is preserved by homomorphisms.) [,m‘ JC 1% A 037 Mama/filmy «J [6+ 46A [75 Ail/amt Than 4" =4 79/‘ game Wei/five W 6/“ fl, {0 W: low “2%) = warmtfms/ofl): 0;. \yfmknwxmw"fld W‘.’ 4 HIM; " WW MM fa) {a 4://W+ a é, 2. Let R be an integral domain, and let a, b E R. Show that a and b are associates if and only if a I b and b | a. (a?) ASW’W/ 61 :3 M #504412 070 W4 4:5“ 7gp 504% um? (46R, W5 5/4, A150 b=au"/ an all. ($1) Asa/M 4/4 MJ TAM 5:4“ fir 50M Mé/g ’ 4M! 'Pév 70% 50M6 V6IQ. F/vm fl“? {7" 13 4/54/ w he “50 m M, 4% no M w M, Ifibaflraa/Jémfl/flltmo/W’é/fzwyw Mmfldeé, 55 455on EM: 5% nng 0. MW alga:va (4a)v:4'(uv)/ 445/ «4‘0, {fax R 13" 4/1 {A7} r2! ngaf/l, u/a am L153 7410 Mu/fV/féqfive Wool/4ith [M 7% cam/Me uv=/. {We R (5 Ram/lulu 253/0, €70 04/569 um hot/a Va :4 4/54 54 EM a M61 V W Wm] 6” 4W? (PAM/M U [5 a [A A} 5% A '5 4/1 dffflcld/Z flc é; 3. Letp=X4—3X3—4X2—l—19X—7andq=X3—4X2+2X+7in (@[X]. (a) Use the Euclidean algorithm to compute the greatest common di— visor of p and q. (You may want to use the back of this page for scratch work.) : + ~2X2+M - ‘ XH F i M ( X H fermwa ' XLWE’ +ZXZ+ 7X _ , 1 , «é: —£ X3 —‘I‘)(z+ 22H? 50 (fl 1‘5 4/! mam—4+6 070 I 7‘ > u ,‘6 5g ‘ZX‘i' 'ZX JFWVM/ & 7L mm k W ’2X1+/0H4 X3 *W‘flXi-i’ 2 5 X1'5X+¥ X ~5X+¥ XZ'SX 1L? ' 0‘ (b) Using your answer from part (a), show that p is reducible in Q[X] but does not haveany roots in Q. 7' +ZX «I 7 ~ X X 5 51+? )(‘C 3X’- 4XZ+W~f X“ —5)(3 H‘X‘ ZXM/xz Hm? ZXg’IflX‘Jr/G’X 4545*? 'X‘ +SX'¥ ET 5” f: (XI-5X+})[X‘+ZX-l). By W Wipm twig/4, m m if rim; [n C M 9H7; 2— ‘ 2 _+ 444 W m at mm W $:—lifl, TM ’a :3 Mia/6&2, [M M M fan? [4 fl, 4. Factor the polynomial X 3 + 3X 2 + 3X + 4 into irreducibles in Z5[X Show your work, and justify your answer. (In particular, be sure to ' explain briefly Why each factor is irreducible.) Look gr 4 [/u (7! X=§,//Z/3/‘f. XsO; WWW :Wfl X4; Hamel: {1M X226Q+12+6+LI=3M / l:th om“ X'Z’ X"Z X3+3lztflrq EXM 3X’6 #‘f PM) Mote Mb X216.” / (aw/7‘ ée/lfl [/7 /{=Z/):9‘, X32? W3i¥$0 ' )(zg: 4+3:{2f0 X4: /5+3=(1¢fl X273 MM M M07? in We! ghee ff; ,5} MI); 2/ H’ E? w'lreoittéié/éa ‘1 070 4mm X'Z 1'7 4/” W Mal/1,, lama A W/[MM/k/ Wt dizzer4 / f5 «la/475 I’l‘fejxa‘é/é’z Wm X5 +3)( #3149 = (X—Z (X 2-23) ...
View Full Document

This note was uploaded on 11/03/2010 for the course MATH 262-338-20 taught by Professor Gieseker,d. during the Fall '10 term at UCLA.

Page1 / 4

Midterm 2 - Solutions - 1. Let R be a ring. Recall that an...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online