In text book "A course in abstract algebra" by author Khanna & bhambri" it is given that,
f(x) = 2( x^2) + 2 is irreducible polynomial over Z.
Because they used the definition "let R be an Integral domain with unity then a polynomial f(x) in R[x] of positive degree (i.e. deg ≥ 1) is said to be irreducible polynomial over R if it can not be expressed as product of two polynomials of positive degree. In other words, if whenever f(x) = g(x)• h(x) Then deg g = 0 or deg h = 0.
Here f(x) = 2(x^2) + 2 = 2(x^2 + 1) Clearly deg g = deg(2) = 0. So f(x) is irreducible polynomial over Z by above definition.
But in book "Contemporary abstract algebra" by "Joseph A. Gallian", they used the definition,
"Let D be an integral domain. A polynomial f(x) in D[x] that is neither a zero polynomial nor unit in D[x] is said to be irreducible over D if, whenever f(x) = g(x) • h(x) with, g(x) & h(x) from D[x], then g(x) OR h(x) is unit in D[x]."
Now here f(x) = 2(x^2) + 2 = 2( x^2 + 1)
Clearly neither 2 nor x^2 + 1 in Z[x] are unit. So that f(x) is reducible by definition of book by "Joseph A. Gallian".
So one book says, it is irreducible over Z but other says it is reducible over Z. Please suggest me which one I should prefer? .