Factoring in ℤ _p [x]

Abstract

Factoring in ℤ _p [x]Factoring in ℤ [x], p prime, is interesting, not only for its own sake, but also because it is useful for factoring in ℤ[x]. See Chapters 8 and 13. For example, x⁴ + 3x + 7 can be shown to be irreducible in ℤ[x] (and therefore in ℚ[x]) by showing that it is irreducible mod 2. For if x⁴ + 3x + 7 = a(x)b(x) then a(x)b(x) can be chosen in ℤ[x]; then x⁴ + 3x +7 =a(x)b(x) (mod 2) and x⁴ + 3x + 7 would factor mod 2. But mod 2, x⁴ + 3x + 7 = x⁴ + x + 1, which is easily shown to be irreducible in ℤ₂[x].