emancipezN

2020-11-22

Decide whether $z\left(\sqrt{-5}\right)$ unique factorization domain or not (ring theory)

aprovard

Skilled2020-11-23Added 94 answers

Step 1

To decide whether$Z\left(\sqrt{-5}\right)$ is unique factorization domain or not.

Step 2

Note that For a integral domain R to be unique factorization domain one of the property is:

if$a={p}_{1}\cdot {p}_{2}\cdot {p}_{3}\cdot \dots ..\cdot {p}_{n}$ ,

$a=q1\cdot q2\cdot q3\cdot \dots .\cdot {q}_{m}$

where p and q are irreducible in R then m=n and each$p}_{i$ is associative of some $q}_{j$ .

Step 3

Here note that$46\in Z\left[\sqrt{-5}\right]$ is an non-zero and non-unit element and 46 can be expressed as:

46=2*23, and

$46=(1-3\sqrt{-5})\cdot (1+3\sqrt{-5})$

but 2 is not associative of$(1-3\sqrt{-5}),{\textstyle \phantom{\rule{1em}{0ex}}}\text{or}{\textstyle \phantom{\rule{1em}{0ex}}}(1+3\sqrt{-5})$

Hence,$Z\left[\sqrt{-5}\right]$ is not unique factorization domain.

To decide whether

Step 2

Note that For a integral domain R to be unique factorization domain one of the property is:

if

where p and q are irreducible in R then m=n and each

Step 3

Here note that

46=2*23, and

but 2 is not associative of

Hence,

Jeffrey Jordon

Expert2021-11-11Added 2605 answers

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