Do you mean that $\mathcal{A}$ is a subgroup of $\mathbb{F}_{q^{2n}}^\times$?

@AlexArvanitakis How do these two properties make the map well-defined?

Right, now that I think of it, we can hardly say anything about exterior powers without the notion of the sign of a permutation.

With the formal definition you wrote it looks like you're counting fixed points as half of 2-cycles, is that indeed what you want?

What do you mean by $Aut(\mathbb{C})$, automorphisms preserving which structure?

Yes exactly I'm talking about profinite completion in this sense

q.anisotropic_primes() gives the list of all primes $p$ (including $\infty$) such that $q$ is anisotropic over $\mathbb{Q}_p$. So by Hasse-Minkowski q is anisotropic over $\mathbb{Q}$ if and only if q.anisotropic_primes() is nonempty (note that in python for a list L, "if L" evaluates to False if L is empty and True otherwise).

For $n=2$, wouldn't any subset $X$ that intersects any line in finitely many points (but at least one) be a counterexample? There certainly exist such subsets that are not algebraic hypersurfaces.

@SugataMandal Two quadratic spaces are isomorphic if there is an isomorphism between the underlying vector spaces preserving the quadratic/bilinear form

Where do these particular forms come from?