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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Gauss' lemma


Author: Hwa Tsang Tang
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
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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.


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DOI: https://doi.org/10.1090/S0002-9939-1972-0302638-1
Keywords: Gauss' Lemma, integral domain, content of polynomial, principal prime ideal, primitive polynomial, greatest common divisor, unique factorization domain, integrally closed domain
Article copyright: © Copyright 1972 American Mathematical Society