2010/057) For what positive integers n is the polynomial x^2n + x^n + 1 irreducible?

let f(x) = x^3-1 = (x-1)(x^2+x+1)

let g(x) = x^2+ x + 1

if w is complex cube root of 1 then f(w) = 0 so g(w) = 0 and g(w^2) = 0

now consider p(x,n) = x^2n + x^n + 1

n cannot be of the form 3k+1 or 3k+2 as

p(x=w , 3k+1) = w^2(3k+1) + w^(3k+1) +1 = w^2 + w +1 = 0 as w^3k = 1

p(x=w^2, 3k+1) = w^4(3k+1) + w^2(3k+1) +1 = w + w^2 +1 = 0 as w^3 = 1

so p(x,3k+1) is divisible by x^2+ x + 1

similarly p(x,3k+2) is divisible by x^2+ x + 1

so n cannot have a facor 3k+1 or 3k+ 2

so h has to of the form 3^k ( k >= 1)

replacing x by x+1 and applying Eisenstein crieteria works and shows that x^2n + x^n + 1 is irreducible for all n of the form 3^m.

as a special case n = 1 means x^2 + x + 1 is irreducible

so n = 1 or e^k for k k >= 1 of 3^k ( k >=0)