Gauss’ lemma
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- by Hwa Tsang Tang PDF
- Proc. Amer. Math. Soc. 35 (1972), 372-376 Request permission
Abstract:
Let $f(x)$ be a polynomial in several indeterminates with coefficients in an integral domain R with quotient field K. We prove that the principal ideal generated by f in the polynomial ring $R[x]$ is prime iff f is irreducible over K and ${A^{ - 1}} = R$ where A is the content of f. We also prove that if $f(x)$ is such that ${A^{ - 1}} = R$ and $g(x)$ is a primitive polynomial in the sense that only a unit of R can divide each coefficient of g, then fg will be primitive.References
- Irving Kaplansky, Commutative rings, Allyn and Bacon, Inc., Boston, Mass., 1970. MR 0254021
- Neal H. McCoy, Rings and ideals, Carus Monograph Series, no. 8, Open Court Publishing Co., La Salle, Ill., 1948. MR 0026038, DOI 10.5948/9781614440086
Additional Information
- © Copyright 1972 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 35 (1972), 372-376
- MSC: Primary 13F20
- DOI: https://doi.org/10.1090/S0002-9939-1972-0302638-1
- MathSciNet review: 0302638