Answer to Question #321131 in Abstract Algebra for Deepak

Question

Prove or disprove that the polynomial 21x^3 - 3x^2 + 2x + 9 is irreducible over Z2 , but not over Z3. Justify your answer.

Answer 1

$To\,\,be\,\,reducible\,\,a\,\,polynomial\,\,of\,\,power\,\,3 needs\,\,to\,\,have\,\,a\,\,root\\\mathbb{Z} _2:\\P\left( x \right) =x^3+x^2+1\\P\left( 0 \right) =0+0+1=1\\P\left( 1 \right) =1+1+1=1\\No\,\,roots\Rightarrow irreducible\\\mathbb{Z} _3:\\P\left( x \right) =2x\\P\left( 0 \right) =0\\x=0-a\,\,root\Rightarrow reducible\\True.$