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MATHMAA

# Irreducible-Polynomials

Definition of Irreducible-Polynomial :

A polynomial p(x) $$\epsilon$$ F[x] is said to be irreducible over the field F

if whenever p(x)=a(x)b(x) with a(x), b(x) $$\epsilon$$ F[x] then one of a(x) or

b(x)  has zero degree.

Note:  p(x) $$\epsilon$$ F[x] is irreducible over F $$\Leftrightarrow$$ whenever

p(x) = a(x)b(x) then one of a(x) or b(x) is a constant polynomial .

eg:    1) f(x) =$$x^{2}-2$$ is irreducible over the field Q[x], where Q is the field of rational numbers,   as f(x)=1*$$x^{2}-2$$ , 1 is constant function .

2) f(x)=$$x^{2} -2$$  is not irreducible over R[x], where R is field of Real numbers  , as f(x) =$$x^{2}-2$$=(x-√2)(x+√2), x+√2 , x-√2 both are degree one polynomials of R[x] .