Is R Z2i a eld Is R Z2i an integral domain Justify your an

Is R = Z2[i] a eld? Is R = Z2[i] an integral domain? Justify your answers.

Solution

z2[i]={a+bi/a,b is belongs to z2};

An integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural setting for studying divisibility. In an integral domain the cancellation property holds for multiplication by a nonzero element a, that is, if a 0, an equality ab = ac implies b = c.

A ring (R,+,.) is called a division ring if it forms a group with respect to the operation \'.\'. If that group is abelian then the ring is called a field

Every finite integral domain is a field.