Question

Let a EC Z such that a? EZ and Rea=0. Let N: Z[0] → Z:2H |212. (a) Show that N() NU {0} for all ze Z[a], and that if ry in Za], then N (2) N(y). (b) Show that Z[a] satisfies the ascending chain condition for principal ideals. (e) (Bonus) Show that Z[iV2 is a Euclidean domain. (Hint: there's a proof in the textbook that Z[i] is a Euclidean domain that you can modify.) (d) Show that Ziv5) is not a UFD.

Answer 1

Let a EC Z such that a? EZ and Rea=0. Let N: Z[0] → Z:2H |212. (a) Show that N() NU {0} for all ze Z[a], and that if ry in Za

Au: a) Let at clz and such that a²E2 Rea=0 Let N: Z[<] + Z :x+1212; #xez[x] Let x=ib : BEZ = N(X) = 12 12 = lib12=6²EN i-e, N(X) EN => NG)E NU 103 (:NSNUL03) Let aly in z(x] :x,yt Z[o] Suppose x=ib and yaici Where bicez kly = y=xk; KEZ = ic=ibk =) (=bk & c2=6²k2 = 6²/e2 » NGỒNG) : N(X) = 62 and Mcy)=C2)

b) z [oc] is a principal ideal Os in generated by the Single Element acclz R= z[2] is an generated by a single integral doman and Element = R is a Principal ideal domain. = R is a UFD, we know if R is a UFD then R satistics the Ascending chain on principal ideals. Condition = z[x] Satistics the Ascending chain on Principal ideas (Proved) Condition c) z lire] is a Euclidian domain Proof.. z [iro] = fatirebla,bez} is an integral domain Let 04x zatitzbezci 159] Such that dariab) = 42+263

ii,x=Q+i5 b €Z [156] and oty = c+in9d E zie such that atinb Ctisod +1526) C-nd) (crired) (c-ifad) @C+2bd)+55 iCbc-ad) C2+2d2 actabd (bc-ad) tita c2t2d² c21ad² => x=yqtr klhere Mao og d(01) <dcy) = ? t < c2 +2d² > 2 [152] is Euclidian domain (Proved)

be Let o$x = at isab and oty =c+ised E Z [iro] dG): a? +262 - 0 Ny - Cat i 5b)(C+isod) - Cac-abd) tih (ad tbc) dGxy) = (ac-2bd)?+ 2cadbc) - Q?c?+4bd2-4abcd + 2(a2d2+2abcd + b²(2) = 42C2+ ub?d? uabcd +292 datuabad +26362 =a²c2+2b2c2 tub²d² trazd? 2 =c? Ca2+263 +2d2(a2+262) Ca2+263) (C2+2d2) — @ From @ and dGx) = doxy)

d) 2 (irs] is not UFD Proof: - 7x3+21= ut se j) (4-55 ;) Where (utbi) €Z (ife] Alu-S5i) EZ [is] But z is Not Associate to wtsi on 4-5i ie, Factorisation of 21 is Not unique in Z Gifs] =) Z [irs] is not UFD.