Chazelle B., Goodman J.E., Pollack R. (eds.)'s Advances in Discrete and Computational Geometry PDF

By Chazelle B., Goodman J.E., Pollack R. (eds.)

ISBN-10: 0821806742

ISBN-13: 9780821806746

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I = Rp with p irreducible. 1, R/I is a field if and only if I is a maximal ideal of R. But I = Ra for some element a of R, and it is clear that Ra is maximal precisely when a is irreducible. 12. 12 29 Residue rings of polynomial rings Let F[X] be the polynomial ring over a field F, and let / = /o + fxX + ■ ■ ■ + fn-iXn~l +Xn,n = deg(/) > 1 be a monic polynomial in F[X], Our aim is to give an explicit description of the residue ring F[X\/F[X]f that we need in subsequent applications. We show that each element of F[X]/F[X]f can be written as a polynomial 90 + 9ie + 92£2 H 1- 3 n - i « n _ 1 where e is a root of / .

An R-submodule of M is a subset L of M which satisfies the following requirements. SubM 1: 0 6 L. SubM 2: If I, /' € L, then I + I' e L also. SubM 3: HI € L and r £ R, then rl € L also. R-module, since the axioms for addition and scalar multiplication already hold in the larger module M. -submodule. R-submodule of a right module M is defined by making the obvious modification to axiom SubM 3: SubMR 1: HI € L and r £ R, then Ir € L also. Clearly, a submodule of a right /^-module is again a right module.

Deduce that A is a root of f(X) if and only if X - A | f{X). 8 33 Let p e Z b e prime. Arguing directly from Gauss's Lemma, show that the polynomials Xn — p are all irreducible for n > 2. Generalize your argument to a proof of Eisenstein's Criterion. Let R be a Euclidean domain and let a in R be neither a unit nor irreducible. Show that the ring R/Ra contains a nontrivial divisor of 0. Let f(Y) = f(Y + 1) be the polynomial obtained by the change of variable X = Y +1 from f(X) € F[X], F a field.

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Advances in Discrete and Computational Geometry by Chazelle B., Goodman J.E., Pollack R. (eds.)

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