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Results tagged with quadratic-forms
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user 3503
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.
2
votes
Positive primes represented by indefinite binary quadratic form
Consider the quadratic number field $K$ with discriminant $D = pq$,
where $p$ and $q$ are primes $\equiv 1 \bmod 4$ (all results below
hold after a suitable modification also for $p = 2$). By results …
- 32.6k
8
votes
Accepted
primes represented by an indefinite binary quadratic form
Meyer (Über einen Satz von Dirichlet, Crelle 103 (1888)) proved that a primitive
binary quadratic form with nonsquare discriminant represents infinitely many primes that
lie in any given compatible re …
- 32.6k
4
votes
Question about Gauss composition law over PID.
Composition in PIDs $R$ is no problem: all you have to do is replace the ring of integers in your favorite proof by $R$.
Speiser wrote his thesis on the theory of binary quadratic forms in number fie …
- 32.6k
10
votes
Higher Composition Law
I have lecture notes that I'd like to turn into a book one day. I have not yet had time to adapt to my new TeX system, and the drawings done via ps-tricks do not yet come out as planned. In addition, …
- 32.6k
9
votes
Accepted
Must a ring which admits a Euclidean quadratic form be Euclidean?
Following Pete's request, I give the following as a second answer.
Take $R = {\mathbb Z}[\sqrt{34}]$ and $q(x,y) = x^2 - (3+\sqrt{34})xy+2y^2$; observe that the discriminant of $q$ is the fundamenta …
- 32.6k
5
votes
Accepted
Verifying an example in the Geometry of Numbers and Quadratic Forms
Consider the form
$$ Q(x) = 2q(x) = (x_1+x_2)^2 + (x_2+x_3)^2 + \ldots + (x_7+x_1)^2.$$
You have to show that it has Euclidean minimum $\frac74$ attained at $X_1 = x_1+x_2 = \frac12$, ..., $X_7 = x_7 …
- 32.6k
11
votes
Must a ring which admits a Euclidean quadratic form be Euclidean?
This is a comment on the manuscript rather than an answer to your question.
When I read that ${\mathbb Z}$-Euclidean principal binary quadratic forms correspond to norm-Euclidean quadratic orders, my …
- 32.6k
28
votes
Zagier's one-sentence proof of a theorem of Fermat
It's been a while since I read Elsholtz's article, but after doing so I felt none the wiser. Below I have translated Heath-Brown's proof into the language of binary quadratic forms; Zagier's proof loo …
- 32.6k
4
votes
Accepted
Lower bounds for split primes in Real quadratic fields
The following is not a full answer, but perhaps gives you an idea of how to approach the result.
Let us consider the claim
$$ p^{hR} \ge \Big(\frac{D}4\Big) $$
for the smallest noninert prime. I first …
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