(a) Show that 2 is equal to the product of a unit and the square of an irreducible in Z[i). (b) Show that an odd prime p e Z is irreducible in Z[i) if and only if p = 3 mod 4. (Hint: Use previous problem). Note: Z(i) is a PID and therefore is a UFD, so irreducibles in Z[i) are also prime in Z[i). This exercise emphasizes that not every prime number in Z is a prime number in Z[i).
(a) Show that 2 is equal to the product of a unit and the square of an irreducible in Z[i). (b) Show that an odd prime p e Z is irreducible in Z[i) if and only if p = 3 mod 4. (Hint: Use previous problem). Note: Z(i) is a PID and therefore is a UFD, so irreducibles in Z[i) are also prime in Z[i). This exercise emphasizes that not every prime number in Z is a prime number in Z[i).
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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![5. (a) Show that 2 is equal to the product of a unit and the square of an irreducible in Zli).
(b) Show that an odd prime pe Z is irreducible in Z[i] if and only if p = 3 mod 4. (Hint:
Use previous problem).
Note: Z[1] is a PID and therefore is a UFD, so irreducibles in Z[i] are also prime in Z[i].
This exercise emphasizes that not every prime number in Z is a prime number in Z[i).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F744af276-3b7c-49cb-a92c-d6b2708fe35d%2F7fcce470-1e05-46ed-ae59-0b4220cf1b8e%2Frg2vqif_processed.jpeg&w=3840&q=75)
Transcribed Image Text:5. (a) Show that 2 is equal to the product of a unit and the square of an irreducible in Zli).
(b) Show that an odd prime pe Z is irreducible in Z[i] if and only if p = 3 mod 4. (Hint:
Use previous problem).
Note: Z[1] is a PID and therefore is a UFD, so irreducibles in Z[i] are also prime in Z[i].
This exercise emphasizes that not every prime number in Z is a prime number in Z[i).
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