Ring in the field of complex numbers

Hello,

let k be a field. On the set

is defined the addition

and the multiplication

Proof that k[i] is a ring with the relations above.

I have a approach but I am not sure.

For proofing it, I guess I have to show that it is a commutative Group, that the multiplication is associative und the distributive laws.

So I want to start with the first. Showing that it is a Abel's group.

For the group I have to proof the existence of neutral elements, the existence of one inverse element, the associative law and the inner composition "+" must be commutative.

Okay, so far but how do I show this concretely?

thanks

all the best