{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Midterm 2 - Solutions

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

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

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

View Full Document

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

View Full Document
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

{[ snackBarMessage ]}

### 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
Ask a homework question - tutors are online