Recall, index of an algebraic scheme $X$ is defined to be the greatest common divisor of the degrees of the space of zero cycles on $X$. I am interested in examples of hypersurfaces in $\mathbb{P}^n_K$ where $K$ is a padic field (for example $\mathbb{Q}_p$) whose index is greater than $1$. If $n=2$ such examples are not very difficult to construct. In general what is known about hypersurfaces with index greater than $1$? Is there any standard trick to produce examples of such hypersurfaces? Any idea/reference in this direction will be most helpful.
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Here is a set of examples I constructed (see `Stably free modules', Amer. Journal, Vol 107, 1985). I will briefly describe them, since these may not be of your interest. Let $k$ be any field and let $f(x)$ be a degree $p$ polynomial over $k$ for a prime $p$, with $f(0)\neq 0$ and $f(x^{p^{n1}})$ irreducible over $k$ and some $n$. Using such an $f$, you can construct a hypersurface in $\mathbb{P}^n_k$ of index $p$. My main interest in the aforementioned paper was for $k$, rational functions over another field, where such $f$ are easy to construct. 


$f:X\to \mathbb{A}^3_K$
given by $f(u,v) = (au^21,v,uv)$. This maps the degree $2$ zerocycle $Z(au^21,v)$ to a $K$point $(0,0,0)$. On the open complements, this map is an isomorphism. – Jason Starr Apr 6 '13 at 23:07