Field/Qi/Exercise

We consider the set

${}K=\mathbb {Q} \times \mathbb {Q} ={\left\{(a,b)\mid a,b\in \mathbb {Q} \right\}}\,$

with the special elements

$0=(0,0){\text{ and }}1=(1,0),$

the addition

${}(a,b)+(c,d):=(a+c,b+d)\,$

and the multiplication

${}(a,b)\cdot (c,d):=(ac-bd,ad+bc)\,.$

Show that ${}K$ is, with these operations, a field.

Create a solution