Summary
ForR a commutative ring, which may have divisors of zero but which has no idempotents other than zero and one, we consider the problem of unique factorization of a polynomial with coefficients inR. We prove that, if the polynomial is separable, then such a unique factorization exists. We also define a Legendre symbol for a separable polynomial and a prime of commutative ring with exactly two idempotents in such a way that the symbols of classical number theory are subsumed. We calculate this symbol forR = Q in two cases where it has classically been of interest, namely quadratic extensions and cyclotomic extensions. We then calculate it in a situation which is new, namely the so called generalized cyclotomic extensions from a paper by S. Beale and D. K. Harrison. We study the Galois theory in the general ring situation and in particular define a category of separable polynomials (this is an extension of a paper by D. K. Harrison and M. Vitulli) and a cohomology theory of separable polynomials.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Beale, S. andHarrison, D. K.,Generalized cyclotomic fields. Ill. J. Math33 (1989), 691–703.
Bourbaki, N.,Commutative algebra. Addison-Wesley, Reading, Mass., 1972.
Chase, S., Harrison, D. K. andRosenberg, A.,Galois theory and Galois cohomology of commutative rings. [Mem. Amer. Math. Soc., No. 52]. A.M.S., Providence, R.I., 1965, pp. 15–33.
Chase, S. andRosenberg, A.,Amitsur cohomology and the Brauer group. [Mem. Amer. Math. Soc., No. 52]. A.M.S., Providence, R.I., 1965, pp. 34–79.
Demeyer, F. andIngram, E.,Separable algebras over commutative rings. [Lecture Notes in Mathematics, Vol. 181]. Springer, Berlin—New York, 1971.
Eckmann, B. andHilton, P. J.,Grouplike structures in general categories. I. Multiplications and comultiplications. Math. Ann.145 (1962), 323–448.
Harrison, D. K. andVitulli, M.,A categorical approach to the theory of equations. To appear in J. Pure Appl. Algebra.
Hungerford, T.,Algebra. Springer, Berlin—New York, 1974.
Janusz, G. J.,Separable algebras over commutative rings. Trans. Amer. Math. Soc.122 (1966), 461–479.
Lang, S.,Algebra. Addison-Wesley, Reading, Mass., 1965.
Magid, A.,The separable Galois theory of commutative rings. Marcel Dekker, New York, 1974.
McDonald, B.,Finite rings with identity. Marcel Dekker, New York, 1974.
Pareigis, B.,Cohomology of groups in arbitrary categories. Proc. Amer. Math. Soc.15 (1964), 803–809.
Samuel, P. andZariski, O.,Commutative algebra. Vol. I. Van Nostrand, Princeton, N.J., 1958.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Harrison, D.K., McKenzie, T. Toward an arithmetic of polynomials. Aeq. Math. 43, 21–37 (1992). https://doi.org/10.1007/BF01840472
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF01840472