1. Employing Gauss' procedure show z^2-4i=0 has a complex root?

Question

Refer fundamental theorem of Algebra

2. If a=b(mod n) , c=d(mod n), then a+c=b+d(mod n) and ac=bd(mod n)
Here = means congruent. I could not find the sign for congruent.

Answer 1

z^2=4i=4 so z=2 z= 2sqrt2+2sqrt2*i and 2sqrt2-2sqrt2*i
2)The necessary and sufficient condition for a=b(mod n) is
a-b=k*n
so if
a-b=k*n and
c-d=k´*n summing
a+c-(b+d)=(k+k´)n Dem.
a=b+k*n
c=d+kn
multiplying ac = bd +H*n so ac-bd=H*n