Real quadratic field Q(sqrt(2))

The two real embeddings send $\sqrt2$ to $\sqrt2$ and $-\sqrt2$. This is a field of rational dimension two; its chosen real embedding does not make it a real vector space.

A rational-linear endomorphism is any two-by-two rational matrix in the basis $1,\alpha$, where $\alpha$ is the displayed quadratic generator. A unit-preserving rational-algebra map must send $\alpha$ to a root of its defining polynomial, so there are precisely the identity and conjugation. The invertible linear maps form $\mathrm{GL}_2(\mathbb Q)$, a much larger group.