Abel's irreducibility theorem explained

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 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 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]}

Corollaries of the theorem include:^[2]

• 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 two different irreducible monic polynomials, then they share no roots.

External links

• Larry Freeman. Fermat's Last Theorem blog: Abel's Lemmas on Irreducibility. September 4, 2008.

Notes and References

1. .
2. .
3. This theorem, for minimal polynomials rather than irreducible polynomials more generally, is Lemma 4.1.3 of . 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. .