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The following is an exercise in Lang's Algebra, Chapter 5, #28.

Let $f$ be a homogeneous degree 2 polynomial in $n$ variables over a field $k$. If $f$ has a non-trivial zero in an extension of odd degree then $f$ has a non-trivial zero in $k$.

Ewan Delanoy gave an answer here for $n=2$ case. I just want to see this result for $n=3$, using the $n=2$ case. For this consider $f(x,y,z)\in k[x,y,z]$ homogeneous of degree $2$, and without loss of generality $f(x,y,1)\in k[x,y]$ has a nontrivial zero in $L^2$, where $\mid L:k\mid$ is odd. Cannot apply induction ($n=2$ case) immediately since $f(x,y,1)$ may not be homogeneous. How can I settle this? Thanks.

user371231
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  • Does this answer your question? If $[L:k]$ is odd, $f$ quadratic form over $k$, then $\exists$ zero in $k$ when $\exists$ zero in $L$ – GreginGre Nov 01 '21 at 08:20
  • @GreginGre I have referred this question in my link. I want to extend the solution of Delanoy. – user371231 Nov 01 '21 at 09:29
  • Looking at the proof of the exercise in its full generality, it looks like the case $n=2$ is just an exception, because you can reduce it to polynomials in one variable, and in that setting it is very easy to use Galois theory, but the proof of the result is done using induction in $[L:k]$. – Aitor Iribar Lopez Nov 02 '21 at 17:06
  • Sorry to be the bearer of bad news, but I don't think that you can really extend this particular solution to the $n>2$ case without significant extra work. What you would love to do to proceed by induction is to show that given a solution over $L$, the coordinates satisfy a linear relation over $k$ and then you could use this linear relation to reduce the number of variables. But there's no obvious reason for there to be a linear dependence relation among the coordinates (at least to me right now). I'd recommend reading the solution by Pete Clark instead, which is short and good. – KReiser Nov 07 '21 at 21:12

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