MATH 242 Midterm, Spring 2014

Spting Semester 2013-14 MATHEMATICS 242 Match 28, 2014 MIDTERM Prof-H. Abu-Khuzam Time: 70 Minutes 1. Let D be a Euclidean domain and a € D. Prove that d(a)=d(1) © a isaunitinD diode ddib) ¥ be i 1/4 . if is wl a Cj ) bs OXY, So dle) = ddl) (=>) Gu, dd eduiy { 10 points]. 4 = Fa A349 ohare Ce & al atv) ple a (e) dle) 2 dl) fF mininel Se i= apa i So G& ww ¢ 3. Prove that in a principal ideal domain ( PID), if ged (a,b) = 1 and a| be, thena c. { 10 points]. ged Cp bi a} wo} {a= Cartsbh ln Sey 56 ¢ VD => Csyac +Sbe tz Ant . { ie 7 7 ps elec XK alee; 2a | tac Poole ae : { =| CL i) da) Show that-f(x) = 2x' +27x3-18x?-15 _ is irreducible in Q[x]. 51, 3} iv, alee, she ¥ 3*h 18 [ 5 points]. Le y roll yu Eb oa s lpe oven - +O iy aaclue na Eixvens Bo ic OY Se b) Let f(x)=x5-2x?+x+1 € Z,[x]. Factor f(x) into irreducible factors in Z,[x]. [ 7 points]. Jed Loi) teerkl= I ° ' 2 z 7 4 i x a % x Jactky ef fée (2) - ZS Fatt{ =O C2) ve POP | xy | snk | pF aext4 Bed — x2 Daxt - a At | — te ae S26 3 e-arintegral domain with field of quotients F. Show that if T is an integral din such that DcTcF, then F is (isomorphic to the field of quotients of T. { 10 points]. af Fou wet Ae smelled Ged Cutter TT The tt Md Tho 7) ip oe bvekd auch Thet fle be DetTe Fick A . - f ff om Mie + 4- ‘ a i» @ Ft of cyto Ties ig Cee hic Bow 2 Slee a ti Mes ‘a | ie J . . . 4 £0) ante, Ds, vere 6.\Let G a group of order 99. a)-—-Show that G has anormal Sylow 3-subgroup H and a normal Sylow 11-subgroup K (&-|~ 3x \A ' [6 points]. a p ' , 4 " _ . . ae * of Sloe Zseegps = lr 3& 2 Bae & GIL divide [I F a < H Ain BR Ger why j lec. a i dee ngs ma pool pat= ae : { j faa 2 fhe. Shee. S- seb J | a = 3 ; ; } iii i i o9 Y ia ics SS x Ne dq Sota Ie Sbog ps ~ feilie 5 Ketel ar A divide 3 ° 1d tC te hy be oD ly at Gh. Seow jf- 39? 2 = iA. £5 GP y : if 1) i ¥ y Anan Lt a Aint - Jheo ut dey ou for — nt, H SK aw chee a b) Show that G = HK and deduce that G is abelian. [ 7 points]. ae HAK ~> oles [wl & Cl | ic ols) 13 ¥ feed => & (e-) = { a Ad = hc, Peed - | ici Leilel = (eilci= i HAK , So Ge AK WK HAK =r€5 < i , { ssa i scat” 2] Fok St l> va fAxn \ cleapen, ) oe wer f < y } ( / Te D bea ipal idea! domain (PID) and a,b are nonzero elements in D. Let y ¢ Dy isa common multiple of a and b}. (a) Prove that I is an ideal of D. (b) Prove that I contains an element m such that m is a least common multiple of aand b. (ie ifajsand b|s, then ms) { 12 points]. 8Let G be a group-of Order 6 such that G has only two non-singelton conjugacy classes-C(a;) and C(a2). Use the class equation to find the order of the center Z(G). [ 8 points]. 9, Answer TRUE or FALSE only [3 points each] : r bee a, -L-- The polynomial f(x) =2x°-1 in Zs[x] is irreducible over Zs. i isj=t, 'si, biaj= b. --L-- The ascending chain condition ( ACC) holds for ideals in Zs[x]. c.f Ifahasa non-singelton conjugacy class, then [G:N(a)|>1. a Every proper prime ideal ina PID is maximal. e -4--IfGisa finite group such that all conjugacy classes in G are singletons, then G is abelian [15 points].