Question

Exercise 2 (pts 5). Let g() E Z[2]. Prove that g(x) is irreducible over Zx if and only if g() is irreducible as polynomial in Q[o].

Answer 1

A ( let g(x) be isseducible in o LU] => g(x) isa i seducible in za] (it) (As 2 ts &)2 Nil let g(2) Lt) Conversely, sely, let in s gla) be ineducid cale Claim ga is inseducible in ala] at if possible gla) be reducible in a (a]. Lig(x)= A(2) Bla) A(a) B(2) SEE ] als and deg A(2), deg B(x) > 0 ut Mr be a common multiple of the denominators of the coff of A(x) il. M ALW) EX[a]. ut L = g cd of coeff. of MACa] I » MA[a] = Lacx) where the coefficients of aca) are relatively pime and acast t [x]. similarly, we get integers (non zero) M' and I such that. . M'B(x) = L bcx) with coefficients of bea) being relatively prime lit M = M/L r = M/L (1) oo g(x) = Mr acx ) b(x) where

Ut ga) bla) t 7 [a 4 say ala) = ao ta, atuta a blan= bot bintant b, na where (ao, ai, gat) = 1 =(bo, e abu) and deg g = it tk an, saya WLOG we can assume urso Claim ur= 1 Hallit ur= l C l mth mto infact a mida o) and (m, l ) = 1 Thus, tam g (a) = ke ala) bla) - (by (1) a gx ) = Cot Ga t .. than then ng mCk &k=0, lygh of m, l) = 11 - - ooo e Ck &k=0,1,. n. C We may assume that a's are relatively prime as we can always take out the common factor and I then deal with g'(x) in g(x)=c'gia where ca gcd (co, u,..,n)) greatest Thus, by above remark, l=1.

13, ut p be a fine such that plme - (ao, .., a = 1 all qo's. Thus not dividing divides each of Similarly, b o Lock bobi bo but P doesn't divide focict with qo such that p a.,aign, ai-1.1 such that I divides. pt bo. boobs bo-1 • Buts now mc. It ao bo to Octoi iti od Pich 10 lotot pot rom ao bo tabi tini e ao bo taitt byta ta 200 ziwy Nylov alid uw wissell As plao, n, plait p divides the teams to the left of a bi similarly, p t ides an the terms to the right of gobo and spm s plm City UTAHU AHTH 23412 AMSHTHOD. tp lai b. = E as ptai pt bo Ad TT2 - НИ where alx) bla) Exx] as g(x ) is ideducible which g(2) = ax bla) is not possible [a] gla) i also Z => i seducible in alx]