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

Norl cas let DE 817 such that x² ET and Rex=0 N! Z Ca] 7 101²; tutto let X -- let b; a) NIN) 1912 = 1ê6 1² = 6² 6 ire. e muro (UNC NU203) aly in 7 lal: ny troldt suppose x = ib ic; where , CEZ aly = y = nk; KEZ let and y= as ic is ac by the & ibk 2) c = b = c 2 = 6² k² 6²1 2 & NOR) | Nly) (NEW) = 62 and N(Y) = (²) [x] principal ideal of generated single element & Eto XlZ. R = 7 [a] is an ning integral domain and generated by single element principal Ideal domath R is UFD, as if R is UFD then R satisfies the ascend ending chain condition on pencopal - Z [a] satis ties the ascending chain condition on principal ideals, (proved) is a a we know a ideals.

Eucludian domain. ºrtides is a Preoob that le+ 2. Let z [iro] satir2b I arbet} in integral domain. o & a- atirabt cives such dlat irab) = a ²+26² 0x = a tirab and oty = ctirad & Civil den) a² + 262 ny = cativ2b) cetired) (ac - 26d) + iva caddbc) diny) cac- abd) ² & alada bc) ² a ² (2 + 462d² - uabed & glad & rabed +62 2² ) aze 2 + 216²d² ryabed tsa²d² Yabed +26² (2 - a²c² & 46²d² + 2a²d² (²(a²+26² ) at ad² Ca²+26²) (a²+26²) (c²+ 2d², C and d! E diniy atirab E ztiva] and oty = ct l'Vad EziCiv at l'12 b (tired cativebsic-ired) (etired) ((-irid) + 26² (2 COM 2 Cuch that y

cac & 26d) & ri (bc- ads c² + 2d² act 2bd et ira C2+2d2 (berad) c² & 2d² 2) u y qt & here - 24 87² € 27 [52] or doo) a dly) c2 + 2d² Euclidian domain (proved

z [iss] is not UFD. Proot 7 x 3 =21=4+ isi) (urusi) ulere (4tr5i) EZ[is] elurusi) E 7 [iro] accociate to utisior yo vei bactorisation ob not unique in z[irs] a zlive] is no 21 i's not a UF DI