Proving $(\mathbb{C},\mathbb{C})$ Is Not A Field

Question

Let's $(\mathbb{C},\mathbb{C})$ be a ordered paired of elements form $\mathbb{C}$ when $\mathbb{C}$ is defined as (a,b).
addition and multiplication is defined as in $\mathbb{C}$.
How do I prove it is not a field if $\mathbb{C}$ is a field

Answer 1

$(2,0)*(0,2)=(0,0)$. Therefore it has zero-divisors. (2,0) does not have an inverse element. ...