Marlow Anderson's A First Course in Abstract Algebra: Rings, Groups, and PDF

By Marlow Anderson

ISBN-10: 1482245523

ISBN-13: 9781482245523

ISBN-10: 1482245531

ISBN-13: 9781482245530

A part of the PWS complicated arithmetic sequence, this article comprises chapters on polynomials and factoring, exact factorization, ring homomorphisms and beliefs, and constructibility difficulties and box extensions.

Show description

Read Online or Download A First Course in Abstract Algebra: Rings, Groups, and Fields PDF

Similar abstract books

Alejandro Adem's Cohomology of finite groups PDF

Adem A. , Milgram R. J. Cohomology of finite teams (Springer, 1994)(ISBN 354057025X)

Get Syzygies and Homotopy Theory PDF

An important invariant of a topological house is its primary workforce. whilst this is often trivial, the ensuing homotopy thought is definitely researched and general. within the normal case, in spite of the fact that, homotopy idea over nontrivial primary teams is way extra frustrating and much much less good understood. Syzygies and Homotopy conception explores the matter of nonsimply attached homotopy within the first nontrivial circumstances and provides, for the 1st time, a scientific rehabilitation of Hilbert's approach to syzygies within the context of non-simply hooked up homotopy idea.

Additional resources for A First Course in Abstract Algebra: Rings, Groups, and Fields

Example text

Try some other representatives of these two residue classes, and see that the same sum is obtained. ✁ It is vitally important that this definition be independent of representatives chosen, for otherwise it would be ambiguous and consequently not of much use. We will shortly prove that this independence of representatives in fact holds. Before we do so, we first observe that we can define multiplication on Zm in a similar way. More succinctly, the definition of the operations on Zm are: [a]m + [b]m = [a + b]m [a]m · [b]m = [a · b]m .

1 Division Theorem for Z Let a, b ∈ Z, with a = 0. Then there exist unique integers q and r (called the quotient and remainder, respectively), with 0 ≤ r < |a|, such that b = aq + r. Proof: We first prove the theorem in case a > 0 and b ≥ 0. To show the existence of q and r in this case, we use induction on b. We must first establish the base case for the induction. You might expect us to check that the theorem holds in case b = 0 (the smallest possible value for b). But actually, we can establish the theorem for all b where b < a; for in this case the quotient is 0 and the remainder is b.

Because r0 = f − gq0 , if e divides f and g, e also divides r0 . Now d is also the gcd of g and r0 obtained from Euclid’s Algorithm. The process is the last n − 1 steps of Euclid’s Algorithm for f and g. So, by the induction hypothesis, because e divides g and r0 , it also divides d. ✷ An important consequence of this theorem is that any two gcds of two given polynomials are just scalar multiples of one another. ✄ Quick Exercise. Show that if e and d are two gcds of polynomials f and g, then d and e are scalar multiples of one another.

Download PDF sample

A First Course in Abstract Algebra: Rings, Groups, and Fields by Marlow Anderson


by William
4.2

Rated 4.05 of 5 – based on 20 votes