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Schönemann's Theorem

If the integral coefficients C_0, C_1, ..., C_(N-1) of the polynomial

f(x)=C_0+C_1x+C_2x^2+...+C_(N-1)x^(N-1)+x^N

are divisible by a prime number p, while the free term C_0 is not divisible by p^2, then f(x) is irreducible in the natural rationality domain.

See also

Abel's Irreducibility Theorem, Abel's Lemma, Gauss's Polynomial Theorem, Kronecker's Polynomial Theorem

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References

Dörrie, H. 100 Great Problems of Elementary Mathematics: Their History and Solutions. New York: Dover, p. 118, 1965.Schönemann, T. "Grundzüge einer allgemeinen Theorie der höhern Congruenzen, deren Modul eine reelle Primzahl ist." Jahresbericht über das vereinigte alt- und neustädtsche Gymnasium zu Brandenburg von Michaelis 1842-Ostern 1844. Brandenburg, 50 pp., 1844.Schönemann, T. "Grundzüge einer allgemeinen Theorie der höhern Congruenzen, deren Modul eine reelle Primzahl ist." J. reine angew. Math. 31, 269-325, 1846.

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Schönemann's Theorem

Cite this as:

Weisstein, Eric W. "Schönemann's Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SchoenemannsTheorem.html

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