Let K =Q(i) denote the field of Gaussian numbers.Which of the following elements of OK are irreducible?1+i,3 – 7i,5, 7,12i,-4 + 5i.Note that 10 =(3+ i)(3 – i) = 2 ·5. Is this an example of non-unique factorization in Z[i] ?Give reasons for your answer.

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Let K = Q(i) denote the field of Gaussian numbers. Which of the following elements of OK are irreducible? 1+i, 3 – 7i, 5, 7, 12i, -4 + 5i.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter5: Rings, Integral Domains, And Fields

Section5.2: Integral Domains And Fields

Problem 24E: If a0 in a field F, prove that for every bF the equation ax=b has a unique solution x in F. [Type...

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Question

Let K =
Q(i) denote the field of Gaussian numbers.
Which of the following elements of OK are irreducible?
1+i,
3 – 7i,
5, 7,
12i,
-4 + 5i.
Note that 10 =
(3+ i)(3 – i) = 2 ·5. Is this an example of non-unique factorization in Z[i] ?
Give reasons for your answer.

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