When estimating camera motion from the fundamental matrix, why can translation only be recovered up to scale using epipolar geometry?

I was wondering once you have recovered the fundamental matrix F such that xFx’ = 0. If you try to recover translation and rotation from F, I am told that this is possible only up to scale? Why so?

Can the Weyl orbits of fundamental weights tell us the Cartan matrix?

Let $ \mathfrak{g}$ be a complex semisimple Lie algebra, $ \Delta$ its root system contained in $ \mathfrak{t}^{\vee}$ for a Cartan sub-algebra $ \mathfrak{t}$ of $ \mathfrak{g}$ . Let $ W$ be its Weyl group. Then we (might) have the following short exact sequence of groups: $ $ 1\rightarrow W \rightarrow Aut(\Delta) \rightarrow E \rightarrow 1$ $ where $ Aut(\Delta)\subset GL(\mathfrak{t}^{\vee})$ is the group of symmetries of $ \Delta$ , and $ E$ (should) be the group of symmetries of the associated Dynkin diagram.

It is possible to see how the map $ Aut(\Delta) \rightarrow E$ is defined from the perspective of simple roots (let them be $ \{\alpha_1,\ldots,\alpha_k\}$ ): let $ \sigma\in Aut(\Delta)$ , then $ \{\sigma(\alpha_1),\ldots,\sigma(\alpha_k)\}$ is a set of simple roots with respect to a choice of positive roots which corresponds to its dominant Weyl chamber. By composing $ \sigma$ with the unique Weyl element mapping this chamber to the original dominant Weyl chamber, we get a permutation of $ \{\alpha_1,\ldots,\alpha_k\}$ . One can see that this permutation preserves the Cartan matrix and thus defines an element of $ E$ .

What I would like to know is whether there is an alternative way to define the above map from the perspective of dominant fundamental weights, namely, let $ \{\lambda_1,\ldots,\lambda_k\}$ be the corresponding fundamental weights, then I observe that any element $ \sigma\in Aut(\Delta)$ induces a permutation of the Weyl orbits generated by the $ \lambda_i$ ‘s. So we almost get the desired map except we haven’t checked this permutation preserves the Cartan matrix. This raises the following question:

Can these Weyl orbits themselves (as polyhedra in the Euclidean space) tell us the Cartan matrix?

As an example, $ A_3$ , these orbits consist of two congruent tetrahedra and one octahedron. We know that there is no edge joining the two tetrahedra in the Dynkin diagram. Can we see this fact from their configuration?

The homotopical proof of the fundamental theorem of algebra

I am reading a homotopical proof of the fundamental theorem of algebra, and the end of the proof is as follows:

… the map $ z\mapsto r^nz^n$ (where $ r$ is a positive real number and $ z\in S^1$ ) is homotopic to a constant map $ z\mapsto a_0$ as maps from $ S^1$ to $ \mathbb R^2-0$ . But $ \mathbb R^2-0\cong S^1$ ,…

So does it follow from here that the maps from $ S^1$ to itself given by $ z\mapsto z^n$ and $ z\mapsto c$ for a constant $ c$ are homotopic, and since the fundamental group of $ S^1$ is $ \mathbb Z$ , the first map induces a nonzero map while the constant map induces the zero map which is contradictory?

Field K and integer n such that the etale fundamental group of (GL_n)_K is the profinite completion of GL_n(K)?

Prove or disprove:

Claim: there exists a field $ K$ and an integer $ n$ such that $ \pi_1^{et}((GL_n)_K)$ is isomorphic to the profinite completion of the abstract group $ GL_n(K)$ ?

Note that then $ G_K \cong \widehat{GL_n(K)}/\widehat{\mathbb{Z}}$ . In particular, $ G_K$ must have a (topological) generating set of size the cardinality of $ K$ .