If $[L:k]$ is odd, $f$ quadratic form over $k$, then $\exists$ zero in $k$ when $\exists$ zero in $L$

Question

I need a hint for this 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 $k$.

I guess my trouble is that I am unsure how to deal with the roots of multi-variable polynomials. Most of Lang's Chapter 5 deals with single variable polynomials over a field.

Answer 1

Here is a proof for two variables. So let $\mathbb L$ be an odd-degree extension of $\mathbb K$, and $f(x,y)=Ax^2+Bxy+Cy^2$. If $f$ has a non-trivial zero in ${\mathbb L}^2$, then by homogeneity either $g(x)=f(x,1)$ or $h(y)=f(1,y)$ has a zero in $\mathbb L$. Say it’s $g$. Then $g$ has a root $\alpha\in {\mathbb L}$. The degree $[{\mathbb K}(\alpha):{\mathbb K}]$ divides the odd number $[{\mathbb L}:{\mathbb K}]$ and is at most $2$, so $\alpha \in \mathbb K$ ; then $(\alpha,1)$ is a non-trivial zero for $f$ in $\mathbb K$.

Answer 2

Lang has been known to cloak theorems in the form of exercises, and this is an instance of this. What you are asking about is an elementary but important result in quadratic form theory, called Springer's Theorem.

(Beware: there is at least one other important result in this area called "Springer's Theorem". The other Springer's Theorem is Theorem 7 in these notes: it's a bit more technical.)

A quadratic form $q(x_1,\ldots,x_n)$ over a field $K$ is called anisotropic if for any vector $v \in K^n$ other than the zero vector, $q(v) \neq 0$. Then the contrapositive, hence equivalent, form of Lang's exercise is:

Theorem (Springer): Let $q$ be an anisotropic quadratic form over $K$, and let $L/K$ be an odd degree field extension. Then viewed as a quadratic form over $L$, $q$ remains anisotropic.

A proof of Springer's Theorem appears (e.g.) in $\S 3$ of these notes on quadratic forms. Despite the fact that this is almost 40 pages into a systematic study of the algebraic theory of quadratic forms, the proof should be self-contained provided the terminology is explained, which I hope I have done above.

Remark: A purely field theoretic result which can be viewed as a special case of Springer's Theorem is: if $K$ is a formally real field -- i.e., $-1$ is not a sum of squares -- and $L/K$ is a finite extension of odd degre, then $L$ is also formally real. (Indeed, a field is formally real iff for all $n$, the quadratic form $x_1^2 + \ldots + x_n^2$ is anisotropic.) This result appears in $\S 15.3$ of my field theory notes, and indeed the proof is essentially the same as the proof of Springer's Theorem.