How many non-isomorphic associative algebras of dimension 2 over the field F\_{p^k} are there?
As much as I have searched, I have not found any results that answer my question; not even for k = 1,2.



How many non-isomorphic associative algebras of dimension 2 over the field F_{p^k} are there?
The answer is: 8 isomorphism classes. The classification up to absolute isomorphism yields: 7.


To show this in a greater and more natural generality, let in general $K$ be a field. I claim that there are 7 isomorphism classes of 2-dimensional $K$-algebras that are not (commutative) fields. Denote by $0\_i$ the $i$-dimensional vector space endowed with null product. The isomorphism classes are represented by the $K$-algebras with basis $(x,y)$ for which the nonzero products among $xx,xy,yx,yy$ are described by


$W\_1$: $xx=x$, $xy=y$;


$W\_2$: opposite to $W\_1$ ($xx=x$, $yx=y$);


(the remaining ones are commutative)


$W\_3$: $0\_2$;


$W\_4$: $K\times 0\_1$: $xx=x$;


$W\_5$: $xx=y$;


(the remaining ones are unitary)


$W\_6$: $xx=x$, $xy=yx=y$ ($K[t]/t^2$);


$W\_7$: $xx=x$, $yy=y$ ($K\times K$).




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This being granted (see below for a proof), it remains to classify quadratic extensions of $K$. If $K$ has characteristic $\neq 2$, they are classified by $K^\*/K^{\*2}\smallsetminus\{1\}$, which has order 2 when $K$ is finite. [This corresponds to twisted forms of $W\_7$, if we include $W\_7$ itself].


If $K$ has characteristic $2$, the separable quadratic extensions are classified by the cokernel of the additive endomorphism $t\mapsto t+t^2$ of $K$ (minus $\{0\}$), and again this cokernel has order 2, and again, including $W\_7$, this are forms of $W\_7$.


This covers finite fields. If we are interested in non-perfect fields of characteristic 2, we get the inseparable quadratic extensions, which are twisted forms of $W\_6$. Including $W\_6$, these are classified by the orbits of $K^{\*2}\rtimes K^{\*2}$ on $K$, acting by homotheties.




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It remains to prove the classification above $W\_{1\le i\le 7}$. Let $A$ be a 2-dimensional associative algebra.


First, the commutative unital case, which is the most standard (already covered by [the previous answer](https://mathoverflow.net/a/383720/14094)): being artininan, if not local, $A$ is a product of at least 2 local unital commutative algebras, and the only possibility is $W\_7$. If local and not a field, the maximal ideal $M$ has dimension 1, $M^2=0$ and we get $W\_6$.


Second assume it commutative, but not unital. Adding a unit realizes our algebra $A$ as ideal of dimension $3$ in a 3-dimensional unital artinian algebra $A'$. If $A'$ is a product of fields, all its ideals are unital (as algebras, albeit not unital subalgebras). So the decomposition of $A'$ as product of local algebras involves a non-field. So either $A'$ is local, or $A'\simeq K\times K[t]/t^2$. In the second case, the two maximal ideals of $A'$ are $\{0\}\times K[t]/t^2$ (which is unital), and to $W\_4$. Next suppose $A'$ is local. If $x^2=0$ for all $x\in A$, then in characteristic not $2$, we deduce $xy=0$ for all $x,y$ by polarization, and we have $A\simeq W\_3$. If $x^2\neq 0$, then setting $y=x^2$ we see that $A\simeq W\_5$.


(In char. 2 it remains to exclude another possibility, where $x^2=0$ for all $x$. If the product were nonzero, let $z$ be in its image, complete to a basis and rescale to get $xx=zz=0$, $xz=zx=z$, and then $x(xz)-(xx)z=xz-0=z$, contradicting associativity.)


Third and last, assume $A$ not commutative. There's a single non-commutative 2-dimensional Lie algebra up to isomorphism, namely with basis $(x,y)$ such that $[x,y]=y$. Hence (applying this to the associated Lie commutator) we can suppose that $A$ has a basis $(x,y)$ with $xy-yx=y$. Since $A$ is non-commutative and has dimension $\le 2$, the centralizer of every nonzero element is a proper subspace, and hence (using associativity), $x^2$ is a scalar multiple of $x$ for every $x$. So, we can write $xx=ax$, $yy=by$, $yx=cx+dy$, $xy=cx+(d+1)y$.


Then (u) $0=y(yx)-(yy)x=y(cx+dy)-(by)x=c(cx+dy)+dby-b(cx+dy)=c((c-b)x+dy)$.


Assume by contradiction $c\neq 0$. By (a) we deduce $d=c-b=0$. Thus $xx=ax$, $yy=cx$, $xy=cx+y$, $yx=cx$. Then $0=(xy)x-x(yx)=(cx+y)x-x(cx)=yx$, so $cx=0$, contradiction. Hence $c=0$, contradiction.


So $c=0$: $xx=ax$, $yy=by$, $yx=dy$, $xy=(d+1)y$.


Then $0=y(xy)-(yx)y=((d+1)-d)y^2=y^2=by$, so $b=0$.


Next $0=(yx)x-y(x^2)=dyx-y(ax)=(d-a)dy$. So $(d-a)d=0$. Also $0=x(xy)-(x^2)y=(d+1)xy-axy=(d+1-a)(d+1)y$. So $(d+1-a)(d+1)=0$.


If $d=0$, the latter writes $(1-a)=0$, so $a=1$, and $A\simeq W\_1$. Otherwise $d=a$, and the latter writes $d+1=0$, so $d=-1=a$. In this case $A\simeq W\_2$ (change $x$ into $-x$ to get the defining basis of $W\_2$).


Finally the $W\_i$ are pairwise non-isomorphic: we see that $W\_1$ and $W\_2$ are not isomorphic because $x$ is a left unit in $W\_1$ while $W\_2$ has no left unit. The other cases are straightforward.




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PS I don't know if there's a reference for this. However, the complex and real cases are easy to locate in the literature. From the complex case (7 cases), it formally follows that the same classification holds in large enough characteristic for algebraically closed fields. Moreover in all cases $W\_i$ has an automorphism group whose automorphism group has no Galois cohomology, so one can predict beforehand that there's a single form over any perfect subfield. The exception being $K\times K$, whose forms correspond to quadratic separable extensions. Anyway I preferred not invoke Galois cohomology in the arguments above.


