Abel's irreducibility theorem

In mathematics, Abel's irreducibility theorem, a field theory result described in 1829 by Niels Henrik Abel, asserts that if ƒ(x) is a polynomial over the a field F that shares a root with an irreducible polynomial g(x), then ƒ(x) is divisible evenly by g(x) (i.e. ƒ(x) can be factored as g(x)h(x) with h having coefficients in F). In other words, if a polynomial shares at least one root with an irreducible polynomial, it necessarily shares all the roots of the irreducible polynomial.

Corollaries of the theorem include:

• 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 ${\sqrt {2}}$ 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 ƒ(x), other than constant multiples of ƒ(x).
• If ƒ(x) ≠ g(x) are both monic irreducible, they share no roots.

References

• Abel, N. H. "Mémoire sur une classe particulière d'équations résolubles algébriquement." (Note on a particular class of algebraically solvable equations) Journal für die reine und angewandte Mathematik 4, 131–156, 1829.
• Larry Freeman. Fermat's Last Theorem blog: Abel's Lemmas on Irreducibility. September 4, 2008.
• Weisstein, Eric W. "Abel's Irreducibility Theorem". MathWorld.