Let consider the number 4 from Z[−3]Z[\sqrt{-3} ]Z[−3]
4=2∗2=(1+−3)∗(1−−3)4 = 2*2 = (1+\sqrt{-3}) *(1 - \sqrt{-3})4=2∗2=(1+−3)∗(1−−3)
So 4 is not uniquely factorized in Z[−3]Z[\sqrt{-3} ]Z[−3] .
Hence Z[−3]Z[\sqrt{-3} ]Z[−3] is not a U.F.D (Unique Factorization Domain)
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