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# 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 .

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 Web Resource. https://mathworld.wolfram.com/AbelsIrreducibilityTheorem.html