A polynomial is said to be irreducible if it cannot be factored into nontrivial polynomials over the same field. For example, in the field of rational polynomials Q[x] (i.e., polynomials f(x) with rational coefficients), f(x) is said to be irreducible if there do not exist two nonconstant polynomials g(x) and h(x) in x with rational coefficients such that f(x) = g(x) h(x) (Nagell 1951, p. 160). Similarly, in the finite field GF(2), x^2 + x + 1 is irreducible, but x^2 + 1 is not, since (x + 1)(x + 1) = x^2 + 2x + 1 congruent x^2 + 1 (mod 2). Irreducible polynomial checking is implemented in the Wolfram Language as IrreduciblePolynomialQ[poly].
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