Abel's irreducibility theorem

In mathematics, Abel's irreducibility theorem, a field theory result described in 1829 by Niels Henrik Abel, asserts that if &fnof;(x) is a polynomial over a field F that shares a root with a polynomial g(x) that is irreducible over F, then every root of g(x) is a root of &fnof;(x). Equivalently, if &fnof;(x) shares at least one root with g(x) then &fnof; is divisible evenly by g(x), meaning that &fnof;(x) can be factored as g(x)h(x) with h(x) also having coefficients in F.

Corollaries of the theorem include:
 * If &fnof;(x) is irreducible, there is no lower-degree polynomial (other than the zero polynomial) that shares any root with it. For example, x2 &minus; 2 is irreducible over the rational numbers and has $$\sqrt{2}$$ as a root; hence there is no linear or constant polynomial over the rationals having $$\sqrt{2}$$ as a root. Furthermore, there is no same-degree polynomial that shares any roots with &fnof;(x), other than constant multiples of &fnof;(x).
 * If &fnof;(x) &ne; g(x) are two different irreducible monic polynomials, then they share no roots.