(Eisenstein's irreducibility criterion) Assume that p is a prime, I(x) = ao + alX + ..
Chapter 36, Problem 36.22(choose chapter or problem)
(Eisenstein's irreducibility criterion) Assume that p is a prime, I(x) = ao + alX + ... +a"x" E Z[x], pia; (0 =:: i =:: n - 1), p21' ao and p l' a". Then I(x) is irreducible in Z[x].Give an indirect proof of this by justifying each of the following statements.(a) Assume that I(x) = (bo + b1x + ... + b"x")(co + CIX + ... + c,x'). Then p does notdivide both bo and co.(b) But p divides one of bo and co. Assume that p l' bo and p !co.(c) Since p1' c, (why?), not all Cj are divisible by p. Let k be the smallest integer such thatP1'Ck andplcj forO =:: j which is a contradiction.(For applications, see 36.23 and 36.24.)
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