Abel's irreducibility theorem

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]

References

  1. Abel, N. H. (1829), "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, doi:10.1515/crll.1829.4.131.
  2. 2.0 2.1 Dörrie, Heinrich (1965), 100 Great Problems of Elementary Mathematics: Their History and Solution, Courier Dover Publications, p. 120, ISBN 9780486613482.
  3. This theorem, for minimal polynomials rather than irreducible polynomials more generally, is Lemma 4.1.3 of Cox (2012). 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. Cox, David A. (2012), Galois Theory, Pure and Applied Mathematics (2nd ed.), John Wiley & Sons, doi:10.1002/9781118218457, ISBN 978-1-118-07205-9.

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