Returning to mathematics......
Yes it's ironic that the OP has hijacked their own thread
I assert that then $$\mathbb{F} \otimes V_n \simeq \mathbb{F}^n$$
I would like to make a notational quibble (two actually) I regard as important.
I'm just trying to make sense of tensor products so nothing I say here is authoritative, I'm just stumbling around in a dark room as Wiles put it.
What I think you mean to say above is this:
"I assert that then $$\mathbb{F} \otimes_\mathbb F V_n \simeq \mathbb{F}^n$$ as vector spaces." Or do you mean $$\otimes_\mathbb V$$? I'm not actually sure which you mean here.
There are two separate things going on here. First, the statement that this is a vector space isomorphism as opposed to, say, an
algebra homomorphism.
This is key because in the subject of this thread, $$\mathbb C$$ is isomorphic to $$\mathbb R^2$$ as $$\mathbb R$$-vector spaces; but not as algebras, since $$\mathbb R^2$$ has no algebra structure at all. That is in fact the essential difference between $$\mathbb C$$ and $$R^2$$.
An algebra is a vector space where there is a multiplication defined on the vectors that's compatible with the underlying scalar multiplication. The classic case is $$\mathbb C$$ regarded as a vector space over $$\mathbb R$$. It's true that we have a vector space, but of course something much stronger is true as well. We have a multiplication defined on $$\mathbb C$$ that makes $$\mathbb C$$ into a ring (in this case a field); and this multiplication is compatible with scalar multiplication by elements of $$\mathbb R$$.
For example $$\alpha (z + w) = \alpha z + \alpha w$$, and this is true whether $$\alpha $$ is real or complex; and whether we regard this as scalar multiplication by a real; or multiplication by a complex number that happens to be real.
It's this extra algebra structure that makes the complex numbers different from the reals.
https://en.wikipedia.org/wiki/Algebra_over_a_field
So first, it's important to say whether your isomorphism is of vector spaces, algebras, or something else (rings, modules ...).
The second missing piece of notation is a subscript on the tensor product sign that identifies the underlying field (or ring). This is important because we are interested in extension of scalars, which is when we take an $$\mathbb R$$-vector space and use a tensor product to make it into a $$\mathbb C$$-vector space. In this case saying $$X \otimes_\mathbb R Y$$ is very different than $$X \otimes_\mathbb C Y$$
Along these lines I wanted to mention that in the very first post of this thread, we have:
It's well known that the complex plane $$ \mathbb C $$ is isomorphic to $$ \mathbb R^2 $$
In fact this is not well known to me! Are these two objects isomorphic as Abelian groups? Yes. As rings? No. $$\mathbb R $$-vector spaces? Yes. $$\mathbb R $$-algebras? No. $$\mathbb C$$-vector spaces or algebras? No.
It is important to be clear about the type of isomorphism (between what type of objects) and if there's a base field involved, what is it?
