Abel's Irreducibility Theorem

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, Schoenemann's Theorem

References

Abel, N. H. ``Mémoire sur une classe particulière d'équations résolubles algébriquement.'' J. reine angew. Math. 4, 1829.

Dörrie, H. 100 Great Problems of Elementary Mathematics: Their History and Solutions. New York: Dover, p. 120, 1965.