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Results tagged with quadratic-forms
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user 297
Algebraic and geometric theory of quadratic forms and symmetric bilinear forms, e.g., values attained by quadratic forms, isotropic subspaces, the Witt ring, invariants of quadratic forms, the discriminant and Clifford algebra of a quadratic form, Pfister forms, automorphisms of quadratic forms.
4
votes
Accepted
Matrix version of number theoretic integral lattice claim
Write $P$ in Smith normal form as $SDT$, where $S$ and $T$ have determinant $\pm 1$ and $D$ is diagonal with diagonal entries $d_1 | d_2 | d_3 | \cdots | d_n$. We may replace $(F, G, P)$ by $(S^{-T} F …
- 156k
1
vote
Matrix version of number theoretic integral lattice claim
A much shorter proof, though one that doesn't explain as much to me. From $P^T F P = r^2 G$, we deduce $P^{-1} = r^{-2} G^{-1} P^T F$ and thus $P^{-1} \in \frac{1}{r^2 \det G} \mathrm{Mat}(\mathbb{Z}) …
8
votes
Accepted
When are rings of the form $K[x_1,...,x_n]/(Q)$ principal ideal domains when $Q$ is quadratic?
PID's have Krull dimension $1$ (or $0$, if you call a field a PID); $A_Q$ will have Krull dimension $n-1$. So the only option is $n=2$ (the case $n=1$ doesn't apply since $k[x]/x^2$ is not a domain).
…
- 156k
8
votes
Accepted
Over which fields is the Sylvester law of inertia valid?
OK, I now have a complete answer; I'll delete the other shortly. The answer to question 2 is yes and easily so; I wonder if I am missing something. If one quadratic form is $\sum a_i x_i^2$, and other …
- 156k
8
votes
Accepted
Fundamental units with norm $-1$ in real quadratic fields
Stevenhagen "The number of real quadratic fields having units of negative norm" Exp Math 1993 makes the following analysis (last paragraph page 127):
Let $D>0$. Let $C$ be the narrow class group of …
- 156k
7
votes
Is the square of the covering radius of an integral lattice/quadratic form always rational?
It's rational. However, I am not sure whether or not the denominator is what you think it is.
Let $\Lambda \subset \mathbb{R}^n$ be your lattice.
The covering radius is the smallest $r$ such that …
- 156k
40
votes
Accepted
Non-diagonalizable complex symmetric matrix
$$\begin{pmatrix} 1 & i \\ i & -1 \end{pmatrix}.$$
How did I find this? Non-diagonalizable means that there is some Jordan block of size greater than $1$. I decided to hunt for something with Jordan …
- 156k