Abel's Irreducibility Theorem

If one root of the equation , which is irreducible over a field , is also a root of the equation  in , then all the roots of the irreducible equation  are roots of . Equivalently,  can be divided by  without a remainder,

where  is also a polynomial over .

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.