Decide whether Z[\sqrt{-5}] unique factorization domain or not (ring theory)

Brennan Flores

Brennan Flores

Answered question

2021-05-21

Decide whether Z[5] unique factorization domain or not (ring theory)

Answer & Explanation

2k1enyvp

2k1enyvp

Skilled2021-05-22Added 94 answers

Step 1
To decide whether Z[5] 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=p1p2p3....pn,
a=q1q2q3....qm
where p and q are irreducible in R then m=n and each pi is associative of some qj.
Step 3
Here note that 46Z[5] is an non-zero and non-unit element and 46 can be expressed as:
46=223, and
46=(135)(1+35)
but 2 is not associative of (135), or (1+35)
Hence, Z[5] is not unique factorization domain.

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