I've got a question form Gille and Szamuely's "Central Simple Algebras' and it's about vector spaces equipped with tensors of certain types.
Let $V$ be a $k$-vector space. For a field extension $K/k$, let $V_K= V\otimes_k K$. Define a tensor $\Phi$ of type $(p,q)$ as an element of $V^{\otimes p}\otimes_k (V^*)^{\otimes q}$ where $p$ and $q$ are non-negative integers and $V^*={\rm Hom}_k(V,k)$ is the dual space of $V$. We let $\Phi_K=\Phi\otimes 1$ be the tensor induced on $V_K$ by $\Phi$.
A $k$-isomorphism between pairs $(V,\Phi)$ and $(W,\Psi)$ of $k$-vector spaces equipped with a tensor of fixed type $(p,q)$ is an isomorphism $f: V\xrightarrow{\sim} W$ of $k$-vector spaces such that $f^{\otimes p}\otimes (f^{\ast-1})^{\otimes q}: V^{\otimes p}\otimes_k (V^\ast)^{\otimes q}\to W^{\otimes p}\otimes_k (W^\ast)^{\otimes q}$ maps $\Phi$ to $\Psi$.
Let $K/k$ be a finite Galois extension. Given a $k$-automorphism $\sigma: K\to K$, tensor by $V$ to give a $K$-automorphism $V_K \to V_K$ which we again denote by $\sigma$. Each $K$-linear map $f: V_K\to W_K$ induces a map $\sigma(f): V_K\to W_K$ defined by $\sigma(f)=\sigma\circ f\circ \sigma^{-1}$.
How do I show that if $f$ is a $K$-isomorphism from $(V_K,\Phi_K)$ to $(W_K, \Psi_K)$ then $\sigma(f)$ is also?
Many thanks in advance!
Of course, no. All non-degenerate symmentic bilinear forms over $\mathbb{C}$ are the same. But not over $\mathbb{R}$.