Let $K$ be a field of characteristic $2$ ($2$ is very important in the statement -- otherwise I can do it myself :) ). On the set $K \times K$ we define the following equivalent relation: $(a, b) \equiv (a', b') $ if and only if there exists a pair $(q, \alpha) \in K^* \times K$ such that: $$ a = q^2 a' + \alpha^2 - b \alpha \quad {\rm and} \quad b = q b' $$

Can you compute explicitely the quotient set $K\times K/\equiv $? Or, if you prefere, can you give a set of representatives for the relation $\equiv$.