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10.22. Let S be {a + b/3i : a, b e Z}, a subring of C. Show that 1+ V3i is irreducible, but not prime.

10.22. Let S be {a + b/3i : a, b e Z}, a subring of C. Show that 1+ V3i is irreducible, but not 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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Could you explain how to show 10.22 in great detail? I also included a list of theorems and definitions in my textbook as a reference. Really appreciate your help!

Let us discuss another sort of integral domain with a nice property. Definition 10.9. A principal ideal domain (or PID) is an integral domain in which every ideal is principal. A field F is an obvious example of a PID; indeed, its only ideals are (0) and F = (1). But we can obtain others through the following theorem. Theorem 10.8. Every Euclidean domain is a PID. Theorem 10.9. Let R be a PID. Suppose that R has ideals Ik, k e N, such that IC ½C I3 C ... Then there exists a positive integer n such that Ik = In for all k > n. Definition 10.10. Let R be an integral domain. Then an element p of R is prime if it is not zero, not a unit, and if p|ab, with a, b e R, then p|a or p|b. Lemma 10.2. Let R be an integral domain, and take 0 + p e R. Then p is prime if and only if (p) is a prime ideal. Definition 10.11. Let R be an integral domain, and take p e R. We say that p is irreducible if it is not zero, not a unit, and if p = ab, with a, b e R, then either a or b must be a unit. Theorem 10.10. Let R be an integral domain. Then every prime in R is irreducible. Theorem 10.11. Let R be a PID and p e R. Then p is prime if and only if p is irreducible.
Transcribed Image Text:Let us discuss another sort of integral domain with a nice property. Definition 10.9. A principal ideal domain (or PID) is an integral domain in which every ideal is principal. A field F is an obvious example of a PID; indeed, its only ideals are (0) and F = (1). But we can obtain others through the following theorem. Theorem 10.8. Every Euclidean domain is a PID. Theorem 10.9. Let R be a PID. Suppose that R has ideals Ik, k e N, such that IC ½C I3 C ... Then there exists a positive integer n such that Ik = In for all k > n. Definition 10.10. Let R be an integral domain. Then an element p of R is prime if it is not zero, not a unit, and if p|ab, with a, b e R, then p|a or p|b. Lemma 10.2. Let R be an integral domain, and take 0 + p e R. Then p is prime if and only if (p) is a prime ideal. Definition 10.11. Let R be an integral domain, and take p e R. We say that p is irreducible if it is not zero, not a unit, and if p = ab, with a, b e R, then either a or b must be a unit. Theorem 10.10. Let R be an integral domain. Then every prime in R is irreducible. Theorem 10.11. Let R be a PID and p e R. Then p is prime if and only if p is irreducible.
10.22. Let S be {a + b/3i : a, b e Z}, a subring of C. Show that 1 + V3i is irreducible, but not prime.
Transcribed Image Text:10.22. Let S be {a + b/3i : a, b e Z}, a subring of C. Show that 1 + V3i is irreducible, but not prime.
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