Abel's irreducibility theorem
Template:Short description In mathematics, Abel's irreducibility theorem, a field theory result described in 1829 by Niels Henrik Abel,[1] asserts that if f(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 f(x). Equivalently, if f(x) shares at least one root with g(x) then f is divisible evenly by g(x), meaning that f(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 f(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 f(x), other than constant multiples of f(x).
- If f(x) ≠ g(x) are two different irreducible monic polynomials, then they share no roots.
References
External links
- Larry Freeman. Fermat's Last Theorem blog: Abel's Lemmas on Irreducibility. September 4, 2008.
- Template:MathWorld
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- ↑ Template:Citation.
- ↑ 2.0 2.1 Template:Citation.
- ↑ This theorem, for minimal polynomials rather than irreducible polynomials more generally, is Lemma 4.1.3 of Template:Harvtxt. 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. Template:Citation.