By Marlow Anderson
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.
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Additional resources for A First Course in Abstract Algebra: Rings, Groups, and Fields
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.
A First Course in Abstract Algebra: Rings, Groups, and Fields by Marlow Anderson