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Abel's Irreducibility Theorem

Abel's irreducibility theorem states that if one root of the equation f(x)=0, which is irreducible over a field K, is also a root of the equation F(x)=0 in K, then all the roots of the irreducible equation f(x)=0 are roots of F(x)=0. Equivalently, F(x) can be divided by f(x) without a remainder,

F(x)=f(x)F_1(x),

where F_1(x) is also a polynomial over K.

See also

Abel's Lemma, Kronecker's Polynomial Theorem, Schönemann's Theorem

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References

Abel, N. H. "Mémoire sur une classe particulière d'équations résolubles algébriquement." J. reine angew. Math. 4, 131-156, 1829. Reprinted as Ch. 25 in Abel, N. H. Oeuvres complètes, tome 1. J. Gabay, pp. 478-507, 1992.Dörrie, H. 100 Great Problems of Elementary Mathematics: Their History and Solutions. New York: Dover, p. 120, 1965.

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Abel's Irreducibility Theorem

Cite this as:

Weisstein, Eric W. "Abel's Irreducibility Theorem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/AbelsIrreducibilityTheorem.html

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