I'd like to know why all quadratic forms satisfy the following:
1) If $F$ is an algebraically closed field, for example, the field of complex numbers, and $(V, q)$ is a quadratic space of dimension at least two, then it is isotropic.
And:
2) If $F$ is the field $Q_p$ of p-adic numbers and $(V, q)$ is a quadratic space of dimension at least five, then it is isotropic.
Any insight on these two is greatly appreciated.
On the first one, from Cassels. He has characteristic not two. He does not say that all forms can be diagonalised over a field, he says that there is a normal basis. We use the polarization identity to get an inner product $\phi,$ a normal basis has $\phi(u_i, u_j) = 0$ when $i \neq j.$ Therefore the form applied to a vector becomes $\sum a_i x_i^2.$ Over an algebraically closed field, we can take all but the first two $x_i$ zero, then take $x_2 = 1,$ and solve for $x_1$ in $a_1 x_1^2 + a_2 = 0.$
The dimension 5 bit is a longer story. I recommend Cassels because he is focused on the rational numbers and the rational integers, he does isotropy for the $p$-adics in great and understandable detail. This includes tables for the Hilbert norm residue symbol, pages 43 and 44, with which the isotropy results are proved, pages 58-60 primarily. Your question is Lemma 2.7, page 60. The proof is pigeonhole on the $p$-adic squareclasses