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*** Let R be a unique factorization domain. Let p E R. If x is irreducible, then it is prime.***

*** Let R be a unique factorization domain. Let p E R. If x is irreducible, then it is prime.***

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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änd read well. Consider the proof of the theorem below. Justify each step that makes a claim, no matter how trivial. It is available in a Word document for your convenience. ***Let R be a unique factorization domain. Let p E R. If x is irreducible, then it is prime.*** Assume x is irreducible. Assume x|ab for some a, b eR. * a = Pip2 " Pn for somen E N and p'SER where each p is irreducible. * b = q,92 * 4m for somem E N and q,'S ER where each g, is irreducible. %3D XC = ab for some c € R. * XC = P1P2 ** Pn9192 *** ¶m *x is an associate with either some p, or some q, (Maybe both!) Wlog x is an associate with p.. *x = P,u for some u E R*. * xu= P1 = (xu )P2P3 Pn * a = x(u-'paP3" Pn) * x|a *x is prime. a =
Transcribed Image Text:änd read well. Consider the proof of the theorem below. Justify each step that makes a claim, no matter how trivial. It is available in a Word document for your convenience. ***Let R be a unique factorization domain. Let p E R. If x is irreducible, then it is prime.*** Assume x is irreducible. Assume x|ab for some a, b eR. * a = Pip2 " Pn for somen E N and p'SER where each p is irreducible. * b = q,92 * 4m for somem E N and q,'S ER where each g, is irreducible. %3D XC = ab for some c € R. * XC = P1P2 ** Pn9192 *** ¶m *x is an associate with either some p, or some q, (Maybe both!) Wlog x is an associate with p.. *x = P,u for some u E R*. * xu= P1 = (xu )P2P3 Pn * a = x(u-'paP3" Pn) * x|a *x is prime. a =
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