Can you compute the quotient set below?  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 $\equiv$. 
 A: The classification of associative algebras in low dimensions has a long
history, beginning with work of Peirce. A good survey, and a classification of
all nilpotent associative algebras of low dimension, over an arbitrary field, also
of characteristic two, can be found in the paper of W. de Graaf, see http://arxiv.org/abs/1009.5339. I do not know if an "explicit list" of representatives
is possible for the general case. 
A: As I mentioned in a comment above, you want to classify monic polynomials of degree two over a field $K$ of characteristic two under the automorphism group of $K[X]$, which is that of maps $X\mapsto \alpha X+\beta$ with $\alpha$, $\beta\in K$ and $\alpha\neq0$.
If the polynomial is reducible, then it is easy to see that it is in the orbit of $X^2$ or of $X(X+1)$, and these two are not in the same orbit.
Suppose then that the polynomial is $X^2+aX+b$ and that it is irreducible. 
If $a=0$, then the polynomial is $X^2+b$, with $b$ not a square, and two such polynomials with $b$, $b'$ nonsquares are in the same orbit iff they differ in a square.
If $a\neq0$, the change of variable $X\leadsto aX$ shows that the polynomial is in the orbit of one of the form $X^2+X+b$. Let $\wp:t\in K\mapsto t^2+t\in K$. The polynomial $X^2+X+b$ is irreducible iff $b$ is not in the image of $\wp$, and it is in the same orbit as $X^2+X+b'$ iff $b$ and $b'$ differ by an element in the image of $\wp$.
We conclude that the orbits are: (i) two special ones, (ii) one for each nonsquare
of $K$ up to addition of squares and (ii) one for each nonimage of $\wp$ up to addition of elements in the image of $\wp$.
This is as far as you are going to get without being more specific about the field (because the arithmetic of the field enters as soon as you want to describe what is a square and what is not, and what is in the image of $\wp$ and what is not) This has nothing to do with the characteristic, and you would have exactly the same problem over any field: classifying irreducible polynomials of degree two is an arithmetic problem —the only difference is that in odd characteristic you need only care about the function $x\mapsto x^2$ while in even characteristic you have to worry also about the map $\wp$. You'd have a similar outcome if you try to classify cubic polynomials in characteristic three.
A: Thank you Mariano for your stimulation. The best result that can be done is bellow. First some notations:  
If $k^2 \neq k$ we denote by $R \subseteq k \setminus k^2$ a system of representative
for the following relation on $k \setminus k^2$: $d \equiv_1 d'$ if and only if there exists $q \in k^{*}$ such that $d - q^{2} d' \in k^2$. 
$T \subseteq k$ a system of representative
for the for the following relation on $k$: $c \equiv_2 c'$ if and only if there exists $\alpha \in k$ such that $c - c' = \alpha^2 - \alpha$. 
Then we have: (the numbers of types has to be computed for any field $k$ of characteristic $2$ - case by case :(:( )


*

*If $k = k^2$ then the
quotient set $k \times k /\equiv $ is equal to $\{\overline{(0,
0)}, \overline{(0, 1)}\}  \cup  \{\overline{(c, 1)} ~|~ c \in T
\} $. Thus in this case there exists $2 + |T|$ types of isomorphism of
$2$-dimensional algebras namely $k_{(0, 0)}$, $k_{(0, 1)}$ and $k_{(c, 1)}$, for some $c \in T$.

*If $k \neq k^2$ then the
quotient set $k \times k /\equiv $ is equal to $\{\overline{(0,
0)},  \overline{(0, 1)}\}  \cup  \{\overline{(c, 1)} ~|~ c \in T
\} \, \cup \, \{\overline{(d, 0)} ~|~ d \in R
\} $. Thus in this case there exists $2 + |T| + |R| $ types of isomorphism of
$2$-dimensional algebras namely $k_{(0, 0)}$, $k_{(0, 1)}$, $k_{(c, 1)}$, $k_{(d, 0)}$,
for some $c \in T$ and $d\in R$.
