Problem 10. Consider the following map between Gaussian integers and Z₂: 6: Z[i]→ Z₂, (a + bi) = a +bmod 2. Explain why is this a ring homormorphism and show that the kernel is the ideal I generated by 1 + i, I = (1 + i). Conclude that I is a maximal ideal.
Problem 10. Consider the following map between Gaussian integers and Z₂: 6: Z[i]→ Z₂, (a + bi) = a +bmod 2. Explain why is this a ring homormorphism and show that the kernel is the ideal I generated by 1 + i, I = (1 + i). Conclude that I is a maximal ideal.
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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![Problem 10. Consider the following map between Gaussian integers and Z₂:
o: Z[i] → Z2, o(a+bi) = a + b mod 2.
Explain why is this a ring homormorphism and show that the kernel is the
ideal I generated by 1 + i, I = (1 + i). Conclude that I is a maximal ideal.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0ab11347-5cf5-45ee-b032-ac50fc9e0f95%2F44525b0c-cbcd-491d-8ad4-5bba99f965b0%2Fc6dohwh_processed.png&w=3840&q=75)
Transcribed Image Text:Problem 10. Consider the following map between Gaussian integers and Z₂:
o: Z[i] → Z2, o(a+bi) = a + b mod 2.
Explain why is this a ring homormorphism and show that the kernel is the
ideal I generated by 1 + i, I = (1 + i). Conclude that I is a maximal ideal.
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