Abstract
This chaptser provides a continuation of Chapter II, §3. We prove standard properties of polynomials. Most readers will be acquainted with some of these properties, especially at the beginning for polynomials in one variable. However, one of our purposes is to show that some of these properties also hold over a commutative ring when properly formulated. The Gauss lemma and the reduction criterion for irreducibility will show the importance of working over rings. Chapter IX will give examples of the importance of working over the integers Z themselves to get universal relations. It happens that certain statements of algebra are universally true. To prove them, one proves them first for elements of a polynomial ring over Z, and then one obtains the statement in arbitrary fields (or commutative rings as the case may be) by specialization. The Cayley-Hamilton theorem of Chapter XV, for instance, can be proved in that way.
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Lang, S. (2002). Polynomials. In: Algebra. Graduate Texts in Mathematics, vol 211. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-0041-0_4
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