# Which implications are there for a quadratic form between being irreducible, non degenerate or smooth?

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?

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 $$n\geq 4$$.