In mathematics, Abel's irreducibility theorem, a field theory result described in 1829 by Niels Henrik Abel,[1] asserts that if ƒ(x) is a polynomial over a fieldF that shares a root with a polynomial g(x) that is irreducible over F, then every root of g(x) is a root of ƒ(x). Equivalently, if ƒ(x) shares at least one root with g(x) then ƒ is divisible evenly by g(x), meaning that ƒ(x) can be factored as g(x)h(x) with h(x) also having coefficients in F.[2][3]
If ƒ(x) is irreducible, there is no lower-degree polynomial (other than the zero polynomial) that shares any root with it. For example, x2 − 2 is irreducible over the rational numbers and has as a root; hence there is no linear or constant polynomial over the rationals having as a root. Furthermore, there is no same-degree polynomial that shares any roots with ƒ(x), other than constant multiples of ƒ(x).
If ƒ(x) ≠ g(x) are two different irreducible monic polynomials, then they share no roots.
^This theorem, for minimal polynomials rather than irreducible polynomials more generally, is Lemma 4.1.3 of Cox (2012). Irreducible polynomials, divided by their leading coefficient, are minimal for their roots (Cox Proposition 4.1.5), and all minimal polynomials are irreducible, so Cox's formulation is equivalent to Abel's. Cox, David A. (2012), Galois Theory, Pure and Applied Mathematics (2nd ed.), John Wiley & Sons, doi:10.1002/9781118218457, ISBN978-1-118-07205-9.