By Martin Kreuzer

ISBN-10: 3540368043

ISBN-13: 9783540368045

Bridges the present hole within the literature among concept and genuine computation of Groebner bases and their functions. A complete advisor to either the idea and perform of computational commutative algebra, excellent to be used as a textbook for graduate or undergraduate scholars. includes tutorials on many topics that complement the fabric.

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Extra resources for Computational Commutative Algebra 1

Example text

X1 5 x1 • Dickson’s Lemma can be generalized to monomial modules as follows. 9. (Structure Theorem for Monomial Modules) Let M ⊆ P r be a monomial module. e. there are finitely many terms t1 , . . , ts ∈ Tn and numbers γ1 , . . , γs ∈ {1, . . , r} such that we have M = t1 eγ1 , .

A) Prove that, given non-associated irreducible elements a, b ∈ R \ {0} , we have (a) ∩ (b) = (ab) . Hint: Let (a) ∩ (b) = (c) , let ab = rc , and let c = sa . Show that s cannot be a unit. Then deduce that r has to be a unit. 2 Unique Factorization b) Use a) to prove that two factorizations of any element are the same up to order and units. c) Conclude that R is a factorial domain. √ Exercise√7. Consider the ring R = Z[ −5] . a. √ Hint: Show that both 2 and 1 + −5 are common divisors. Exercise 8.

Furthermore, let m ≥ 2 and f1 , . . , fm ∈ R \ {0} . a) Let f1 = c1 p∈P pαp and f2 = c2 p∈P pβp be factorizations of f1 and f2 with units c1 , c2 ∈ R , with αp , βp ∈ N, and with αp = βp = 0 for all but finitely many p ∈ P . Then the element pmin{αp ,βp } gcd(f1 , f2 ) = p∈P is called a greatest common divisor of f1 and f2 , and the element pmax{αp ,βp } lcm(f1 , f2 ) = p∈P is called a least common multiple of f1 and f2 . b) If gcd(f1 , f2 ) = 1 , we say that f1 , f2 are coprime or relatively prime.

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Computational Commutative Algebra 1 by Martin Kreuzer

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