Let $q(x_1,x_2,...,x_n) =\sum a_{ij}x_ix_j\in k[x_1,x_2,...,x_n] \; (a_{ij}=a_{ji})$ be a quadratic form in $n$ variables over a field $k$ of characteristic $\neq 2$, but not assumed algebraically closed nor anything else.
Question I
Are there any implications between the following properties?
a) The polynomial $q$ is irreducible i.e. it is not the product of two linear forms in $k[x_1,x_2,...,x_n]$.
b) The quadratic form $q$ is nondegenerate i.e. its rank is $n$ [the rank of $q$ being the rank of the matrix $(a_{ij})$] .
c) The quadric $V(q)\subset \mathbb P^{n-1}$ given by the equation $q=0$ is smooth.
Concretely c) means that the linear system of $n$ equations $\frac {\partial q} {\partial x_i}(a)=0$ has no solution $a\in k^n$ besides the solution $a=0$.
Question II
Is there a book in which these implications are spelled out?


About question I

  1. For $n=1$ all nonzero quadratic forms are nondegenerate, smooth and reducible.
  2. For all $n\geq 1$ the properties "nondegenerate" and "smooth" are equivalent.
    Quadratic forms satisfying these conditions are often called regular in the literature.
  3. For $n\geq3$ these properties "nondegenerate" and "smooth" imply "irreducible" .
  4. For $n=2$ the property "nondegenerate" (or "smooth") does not imply "irreducible".
    For example the form $x_1x_2\in k[x_1,x_2]$ is non degenerate but of course the polynomial $x_1x_2$ is reducible.
  5. However for $n=2$ "irreducible" $\implies$ "nondegenerate".
  6. For $n=3$ the implication "irreducible $\implies$ "nondegenerate" is false in general.
    For example the quadratic polynomial $x_1^2+x_2^2\in \mathbb R[x_1,x_2,x_3]$ is irreducible but the corresponding quadratic form is degenerate.
    On the other hand the implication "irreducible $\implies$ "nondegenerate" is true if $k$ is algebraically closed.
  7. For all $n\geq 4$ and all fields $k$ the implication "irreducible" $\implies$ "nondegenerate" is FALSE!
    It suffices to consider the quadratic polynomial $x_1^2+x_2^2+x_3^2 \in k[x_1,x_2,..., x_n]$ : that polynomial is irreducible over any field of characteristic not $2$, but the corresponding quadratic form is of course degenerate as soon as $n\geq 4$.

About question II
No, I don't know any book nor any other document in which these (easy) results are explicitly spelled out. Does some user know one?
Since questions of this type often pop up (see here, here or there ), I decided to try and give a reasonably complete answer.

  • 1
    $\begingroup$ I'm sure that (most of) these results are contained inside The algebraic and geometric theory of quadratic forms. $\endgroup$
    – Eoin
    Sep 22 '20 at 19:39
  • 3
    $\begingroup$ @Eoin. No, they are definitely not. I challenge you to post the 7 page numbers corresponding to my 7 points. Anyway, I think a vague comment that ("most" ?) results are somewhere inside a 450 page book is not especially helpful. $\endgroup$ Sep 22 '20 at 19:47
  • $\begingroup$ 1903 answers and one question. good ratio. You might enjoy Classical Groups and Geometic Algebra by Larry C. Grove. everything relevant is surely in Lam, Quadratic Forms over Fields, but I don't remember answering any question on MSE by quoting from Lam. Also, as to my interest, he does not do spinor genus. He does do the spinor norm as a group homomorphism as Theorem 1.13 on page 108, in the chapter on Clifford algebras. Does not define genus, I guess that is what happens when the coefficients are in a field..... $\endgroup$
    – Will Jagy
    Sep 22 '20 at 20:07
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    $\begingroup$ Thanks for your comment, @Will Jagy. I checked in Lam (whom I greatly admire) but could find nothing related to my question. Could you please give a precise reference? $\endgroup$ Sep 22 '20 at 20:32
  • 1
    $\begingroup$ @Georges No, of course the post of mine that you cite is in a number of books or notes on topology. I also don't know what information the OP has to ask this question; I was only providing a reference that the OP could look to if they were trying to find more about the subject and your examples. Lam's book was another that I thought could be a good reference. (Both of these though are really just comments). $\endgroup$
    – Eoin
    Sep 22 '20 at 23:19

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