A polynomial that is not divisible by any smaller polynomials other than trivial ones is an irreducible polynomial. Let \(f (x)\) be a polynomial
where the coefficients \(f_{0}, \ldots, f_{d}\) are elements of a field F. If there is another polynomial \(g(x)\) over F with degree between 1 and \(d-1\) such that \(g(x)\) divides \(f (x)\), then \(f (x)\) is reducible. Otherwise, \(f (x)\) is irreducible. (Nonzero polynomials of degree 0, i.e., nonzero elements of F, divide every polynomial so are not considered.) As an example, the polynomial \(x^2 + 1\) over the finite field F 2 is reducible since \(x^2 + 1 = (x + 1)^2\), whereas the polynomial \(x^2 + x + 1\) is irreducible.
A representation of the finite field 
can be constructed from a representation of the finite field F q together with an irreducible polynomial of degree d, for any d; the polynomial \(f (x)\) is called the field polynomialfor this field. In the...
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Kaliski, B. (2005). Irreducible Polynomial. In: van Tilborg, H.C.A. (eds) Encyclopedia of Cryptography and Security. Springer, Boston, MA . https://doi.org/10.1007/0-387-23483-7_210
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DOI: https://doi.org/10.1007/0-387-23483-7_210
Publisher Name: Springer, Boston, MA
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