Ring in the field of complex numbers

Hello,

let k be a field. On the set

$\displaystyle k \left[ i \right] := \lbrace\left(x,y\right) \mid x,y \in k\rbrace$

is defined the addition

$\displaystyle \left( x,y \right) + \left( x',y' \right) = \left(x+x', y+y' \right) ~ \forall x,x',y,y'$

and the multiplication

$\displaystyle \left( x,y \right) \cdot \left( x',y' \right) = \left( xx' -yy', xy' + x'y \right) ~ \forall x,x',y,y' \in k$

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