# Abel's irreducibility theorem

• 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 ${\displaystyle {\sqrt {2}}}$ as a root; hence there is no linear or constant polynomial over the rationals having ${\displaystyle {\sqrt {2}}}$ as a root. Furthermore, there is no same-degree polynomial that shares any roots with ƒ(x), other than constant multiples of ƒ(x).