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Results tagged with quadratic-forms
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user 4213
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.
9
votes
Root systems and sums of squares
If a quadratic form in $n$ variables is the sum of the squares of
$n$ integer linear forms, it's the sum of the squares of $n$ rational linear forms.
Thus it's equivalent
as a rational quadratic form …
- 20.8k
3
votes
Accepted
Diagonalization of quadratic forms over euclidean rings
Quadratic forms over $\mathbb{Z}$ don't diagonalize in general.
Even positive definite rank two forms like $3x^2+2xy+5y^2$ can't be diagonalized.
Inverting $2$ won't help things.
- 20.8k
8
votes
Accepted
Does a positive binary quadratic form represent a set of primes possessing a natural density
Accoring to H. Lenstra, Chebotarev's theorem holds both for Dirichlet and
for natural density (but he doesn't give a reference in this document).
Applying Chebotarev to the extension $H/\mathbb{Q}$ wh …
- 20.8k
4
votes
Accepted
Invariant quadratic forms of irreducible representations
There are certainly examples over $k=\mathbb{Q}$ where $\dim T\ge2$.
Let's take the cyclic group $G$ of order $5$ and the representation
space
$$V=\{(a_0,\ldots,a_4)\in\mathbb{Q}^5:a_0+\cdots +a_4=0\} …
- 20.8k