The polynomial f(x) = x^2 + 1 is not irreducible over the finite field F_2. Evaluating the polynomial at the elements of F_2, specifically f(0) = 1 and f(1) = 0, confirms that it has a root in the field. Therefore, f(x) can be factored as (x + 1)(x + 1) in F_2[x]. This conclusion is definitive based on the properties of polynomials over finite fields.
PREREQUISITESMathematics students, particularly those studying abstract algebra, educators teaching polynomial theory, and researchers interested in finite fields and their applications.
Looks good.jimmycricket said:Homework Statement
Is f(x)=x^2+1 irreducible in \mathbb{F}_2[x]
If not then factorise the polynomial.
The Attempt at a Solution
\mathbb{F}_2[x]=\{0,1\}
f(0)=1
f(1)=1+1=0
Hence the polynomial is not irreducible