Question

DUP . 2.2 A formal definition of C[t runs as follows. Consider the set of all infinite sequences (an)neN = (20, 21,..., Ans...) where an EC for all n E N, and such that an = 0 for all but a finite set of n. Define operations of addition and multiplication on S by the rules (an) + (bn) = (un) where Un = an + bn (an) (sn) = (vn) where yn = anbo + an-ıbı + ... + apbn Prove that C[t), so defined, satisfies all of the usual laws of algebra for addition, subtraction, and multiplication. Define the map 0:0 S (k) = (k, 0, 0, 0, ...) and prove that O(C) SS is isomorphic to C. Finally, prove that if we identify a EC with e(a) S and the 'indeterminate 1 with (0,1,0,0,0....) E S. then (an) = 10 + ... + apr", where N is chosen so that an = 0 for n > N. Thus we can define polynomials as sequences of complex numbers corresponding to the coefficients. following algebraic
Galois Theory chapter 2

Answer 1

to the termal definition of e[t] rune as follows', s be the set of all intinite sequenes (9) mei (20, Quant anir.) were an eayneiN and an=0 ter all but a tinite set of n. now (an) + (bn) = (Un) where una antoni [an. (bn) (Un) Una an bot ana bt trait goby aul (an) (bn) & S. Then an, br 64 and a. there erist No anod, NL E IN such that anco y nynt, bnzo &ny N2 Then, (an) + (bn) = (un) cohere a una ant bn t¢ ar una ant bn=0 & my man & NL, NL (am) - f bon) – lam) + (-ba) -- Ren) where a una an-bnt & and uncovny man? NL, NL (an). (bn) = (wow) where who a ambo tan-abitat go bn. Er wn = 0 & ny (NTN2) so Sc[t], bo detied , satisties all the usual lawe of as algebra for addition, substruction and multiplication, d: 8s 0 (k) (6,0,0 - -.) List atib, (tid Ek o latin) & (Cotid)) = o(lar) ti (b+d)) = ((a+el+itd), 0, 0 -- 0) - (atib, 0,0-) + (ctid, o, ow - Olat 6) + (atida a fentib) (ctid))=0 (ſae-bdhti (betad)) D o llac-bd) til betod),0,0 ) = ((atib) (crid), ord) ) ((a+ib), 0,0,- (cetid), 0, 0 - 01Q+i b]] o lectid)) so o home in a homopophism a o now for some aye¢ Ola)= 0 (0%) (0,0,-. ) - 019---) ny o is one-one. de Then ock) is ismerphic & inn an Any aed can be identified as o(a) (ano, ) we identity t with col, o --) ts. are due (an)ES Then there exists NG IN O an = o n Also ait& VIEN, (an) a (ao, alt an, oo . . ] such that glotlQ00,--.0)+ QHO,,--. o) + a2 (0,0,10---0) +-es + ap (0, 0, -0,1,0 --0) Ntaposition NOW tz (0,1,0-0) bowling +7 (01-- :) (02, -") 2(0,0,1,0 , 0) 13-trit (00.2, 0 -- 0) (0,1,0,--0) =(0,0,0,1) 0---0) (0,0,- - Ond, o--) 11 - foto nth placu (and (or otet dot ait tatt.. tautN.. This we eas detine polynomiale ak sequence of complen numbers conreppending to the Coetficients. Scanned with CamScanner CS Scanned with CamScanner