12
votes
Accepted
Brauer group of rational numbers
What reference are you reading?
For a field $K$, every finite-dimensional central simple $K$-algebra $A$ is isomorphic to ${\rm M}_n(D)$ where $n$ is a positive integer and $D$ is a division ring with ...
- 50.6k
12
votes
Accepted
Brauer groups and field extensions
No: the conic $C:X^2+Y^2+1=0$ splits over the field $L=\mathbb{Q}(x)[y]/(x^2+y^2+1)$, since $(X,Y)=(x,y)$ is an $L$-point of $C$. However $L$ has no subfields algebraic over $\mathbb{Q}$ other than $\...
- 4,746
10
votes
Accepted
Reference request: correspondence between central simple algebras and quadratic forms
Everything is in Lam's book Introduction to Quadratic Forms over Fields. Theorem III 5.1 says:
All central simple algebra $A$ of dimension $4$ is quaternion. That is $A \cong \left(\frac{a,b}{k}\...
- 534
10
votes
Reference request: correspondence between central simple algebras and quadratic forms
In the formulation, presumably on the right side what is intended are 3-dimensional non-degenerate quadratic spaces (up to isomorphism), with discriminant 1 (same as $4^3$ mod squares as John Ma notes)...
8
votes
Elementary classification of division rings
The classification is trivial for a $\,C_1$-field $K$ (that is, such that any homogeneous polynomial in $K[x_1,\ldots ,x_n]$ of degree $<n$ has a nontrivial zero): the only such division algebras ...
- 38k
7
votes
Accepted
Example of a central simple algebra
You can take: $F=\mathbb{C}(X_1,Y_1,\ldots,X_n,Y_n)$ and take the tensor product of quaternion algebras $$A=(X_1,Y_1)_F\otimes_F\cdots\otimes (X_n,Y_n)_F.$$ Here $A$ contains a subfield $E$ ...
- 1,766
6
votes
Reference request: correspondence between central simple algebras and quadratic forms
On a conceptual level, I would like to see this as an instance of a sporadic isomorphism of algebraic groups. Take the conjugation action of $GL_2$ on the space of $2\times 2$-matrices with trace 0, ...
- 17.4k
6
votes
units in distinct division algebras over number fields---are they definitely not isomorphic as abstract groups?
I know this is a very old question but I saw it just a few days ago. Super-rigidity can indeed be applied not for "higher rank lattices" but for the global points $D^*/F^*$ (which form a lattice in ...
- 11.2k
5
votes
What is the quotient group $D^*/{F^*(1+P_D)}$ for a quaternion division algebra $D$ over a local field $F$?
To give an alternative answer, let us first recall the article "Construction of Locally Compact Near-Fields from $p$-Adic
Division Algebras" by Detlef Groger:
Fix a prime element $\pi_F$ of ...
- 143
4
votes
Accepted
What is the quotient group $D^*/{F^*(1+P_D)}$ for a quaternion division algebra $D$ over a local field $F$?
Yes, we can.$\newcommand{\order}{\mathcal{O}}$ $\newcommand{\Z}{\mathbb{Z}}$ $\newcommand{\prim}{\mathcal{P}}$ $\newcommand{\F}{\mathbb{F}}$
First, let me remind you of the following explicit ...
- 5,382
3
votes
Accepted
Dimension of maximal tori in division algebras
No, there are no such examples, but I don't know any way to attack this by methods of ring theory. The theory of linear algebraic groups gives a very illuminating insight into this matter, by ...
3
votes
Accepted
When an algebra isomorphism preserves positive involution
The answer to your general question is NO.
Assume for simplicity that $A=B$, and that $A$ is a central simple $K$-algebra. Then $\varphi$ is an inner automorphism by Skolem Noether 's Theorem.
Write $\...
- 1,766
3
votes
Accepted
On Dirac/ Clifford matrices
If you require that the matrix $C$ in \eqref{2} preserves the involutions \eqref{3}, then you get the consequence $\tilde{\gamma}^\mu = C C^* \tilde{\gamma}^\mu (C C^*)^{-1}$, which then implies that $...
- 21.5k
3
votes
Elementary proof that a central simple algebra over a field having a maximal subfield is a cyclic algebra
The proof uses as an essential ingredient Proposition 2.2.8, which itself relies on Lemma 2.2.9 telling you that the k-algebras in $M_n(k)$ isomorphic to $k^n$ are conjugate to the subalgebra of ...
- 1,017
3
votes
Accepted
Are there any central simple algebras admitting a standard basis?
As suggested by @Kimball I develop my comment. An important class of central simple algebras consists of the cyclic algebras: assume that the field $k$ contains a primitive $n$-th root of unity $\zeta ...
- 38k
2
votes
Accepted
Hasse invariant and the Clifford algbera
You can find some information about this in Lam's book "Introduction to quadratic forms over fields," particularly in the 3rd chapter. I'll give the answer in a "field agnostic" ...
- 476
2
votes
Accepted
Crossed product division algebra
Take a prime number $p>2$ and a field $k$ of characteristic zero which does not contain a $p$-th root of unity. Assume that $K/k$ is a Galois extension of order $p$ with a Galois group $\langle g\...
- 5,039
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