Artin's conjecture on forms over the $p$-adic numbers

For a field $k$, say that it is $C_2$ if, for every form $F\in k[X_1,\ldots,X_n]$ of degree $d$ with $n>d^2$, the equation

$$F(X_1,\ldots,X_n)=0$$

has a nontrivial solution in $k^n$. Artin's conjecture. For each prime number $p$, the field $\mathbb{Q}_p$ is $C_2$. The conjecture is false: Terjanian gave a counterexample in degree $d=4$ and $p=2$, later generalized to arbitrary $p$.