The following theorem is often called the fundamental theorem of projective geometry:
Let $k$ be a field and let $n \geq 3$. Let $X$ be the partially ordered set of nonzero proper subspaces of $k^n$. Then every poset automorphism of $X$ is induced by a semi-linear automorphism of $k^n$, i.e. a set map $f:k^n\rightarrow k^n$ for which there exists a field automorphism $\tau:k \rightarrow k$ such that
$$f(c_1 \vec{v}_1+c_2\vec{v}_2) = \tau(c_1) f(\vec{v}_1) + \tau(c_2) f(\vec{v}_2)$$
for all $c_1,c_2 \in k$ and $\vec{v}_1,\vec{v}_2 \in k^n$.
Question: who was the first person to prove this, and where does their proof appear? I know it has its origins in 19th century work of von Staudt, but I don't think that the above theorem appears in his work. On page 52 of Baer's book "Linear Algebra and Projective Geometry", he says that the first proof was due to Kamke, but he does not give a reference.
Best Answer
The version you state is definitely a 20th century development, only marginally related to Von Staudt's theorem. Here is a translation of the relevant section of Karzel & Kroll's Geschichte der Geometrie seit Hilbert, p. 51 (notation should be self-explanatory):
One might add that Darboux (1880, p. 59) already states and attributes the theorem:
He only sketches a proof, and Schur (1881, p. 254) comments:
Further references, giving various versions of the theorem but apparently never tracing it beyond Baer, are Dieudonné (1955, p. 72), Artin (1957, p. 88), Bourbaki (1970, Exerc. II.9.16), Jacobson (1974, p. 470), Samuel (1986, p. 32), Berger (1987, 5.4.8), Bennett (1995, p. 203), Jeffers (2000, p. 810).