Do Gaussian integers form a group under multiplication

Could you please show a full general proof checking for identity, closed, inverse and so on
PLEASE HELP AND SHOW WORKING !!!! :)

to make thing clear, i will assume in the following:

z = a+bi, where a,b are integers
w = c+di, where c,d are integers
u = h+ki, where h,k are integers

strictly speaking, the answer straight-away is no, only the non-zero numbers could possibly qualify, since 0 = 0+0i can not possibly have a an inverse. we will see later, that other elements do not, as well.

first, we check closure:

zw = (a+bi)(c+di) = a(c+di) + (bi)(c+di) = ac + adi + bci + bd(i^2)

= (ac-bd) + (ad+bc)i, wich is in Z[i], since ac-bd and ad+bc are both integers.

next, we check associativity:

(zw)u = [(a+bi)(c+di)](h+ki)

= [(ac-bd) + (ad