12
votes
Accepted
Infinite dimensional finitely generated algebraic division algebra
This is a fairly well known old and open (as far as I know) problem: Kurosh’s Problem for division rings. See, for example, Question 3 in Agata Smoktunowicz’s 2006 ICM talk.
- 35.2k
9
votes
Accepted
Is there a classification of the $p$-adic normed division algebras?
Clearly you are assuming some kind of finite-dimensionality over the center.
To classify the finite-dimensional associative division algebras over $\mathbf Q_p$, or more generally over a local field, ...
- 50.6k
9
votes
Accepted
Octonion algebras over $\mathbb{F}_p(t)$
The complete classification for fields of characteristic unequal to 2 is given in Section 8 of Serre's paper
J.-P. Serre. Cohomologie galoisienne : progrès et problèmes. Séminaire Bourbaki, Volume 36 ...
- 17.4k
8
votes
Accepted
Image of the norm map for degree $3$ galois extension over $\mathbb{Q}$
There is a general strategy to tackle such question. Since the norm is multiplicative, it make sense to look for a prime $p$ not in the image. Then, we can ask about the number of times $p$ divides a ...
- 4,189
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
5
votes
Field of definition of a finite dimensional division algebra and how to reduce it
If $L/K$ is a field extension and $D/K$ is an algebra such that $D \otimes_{K} L = D_L$ is a division algebra, then $D$ is a division algebra. Note that the map
$$d \otimes 1: D_L \rightarrow D_L$$
...
- 51
5
votes
Accepted
Left vs right degree of skew-field extensions
Anything that might happen does happen.
In
Schofield, A. H., Artin’s problem for skew field extensions, Math. Proc. Camb. Philos. Soc. 97, 1-6 (1985). ZBL0574.16008.
it is shown that for any ...
- 35.2k
4
votes
Accepted
Unital nonalternative real division algebras of dimension 8
The answer on math.SE points to a construction that produces an algebra of the desired type. (But maybe not all of them?) Here is a different kind of answer: There are uncountably many isomorphism ...
- 2,178
4
votes
Representations of $SL_1(D),$ where $D$ a division algebra over a local field
The representation theory of ${\rm SL}_1 (D)$ was the topics of Göran Kirchner's PhD (defended in 2007, under the supervision of E.-W. Zink, Berlin). To my knowledge it has not been published.
https:/...
- 6,349
4
votes
Endomorphism algebras of restricted representations
EDIT: Finiteness isn't necessary for this argument. Instead I use that $V$ is semisimple over $N$, $|G:N|<\infty$ and $char(k)=0$.
$Res_N^G(V)$ is semisimple because it is the restriction of a ...
- 9,650
4
votes
Accepted
Moufang identities and Moufang plane
This is discussed and proved in detail in Hall, Marshall jun., The theory of groups, New York: The Macmillan Company. xiii, 434 p. (1959). ZBL0084.02202, specifically in chapter 20 "Group Theory ...
- 5,654
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
Is Hurwitz's theorem true in constructive mathematics?
There is a weakening of Hurwitz's theorem that is true constructively, with essentially the same proof:
Let $A$ be a division composition algebra. Then any chain of proper subalgebras $\mathbb{R} = ...
- 771
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
3
votes
Moufang identities and Moufang plane
The book "Moufang polygons" by Tits and Weiss provides a complete account on the more general question of Moufang $n$-gons (for Moufang planes, $n=3$).
One can read some parts of it to get ...
- 14k
3
votes
Accepted
Modular forms on central division algebra of degree $\ge 3$
For the first question, it is only true that if $D$ is a (totally) definite quaternion algebra over a number field $K$, then the weight 0 automorphic forms factor through a finite set (1-sided ideal ...
- 6,039
3
votes
Moufang identities and Moufang plane
A very accessible book for such connections between geometric and algebraic properties in general, is John Faulkner's "The Role of Nonassociative Algebra in Projective Geometry" (https://...
- 6,614
3
votes
When is the following fraction an integer $\frac{3^a}{2^b-3}[(\frac{2^b}{3})^c - 1]$ where $a,b,c \in \mathbb{Z}$?
The formula can be rewritten as follows:
$$3^{a-1} \frac{(\frac{2^{b}}{3})^c - 1}{\frac{2^b}{3} - 1}$$
Using the general formula $\sum_{i=0}^{n} x^n = \frac{x^{n+1}-1}{x-1}$, this rewrites to
$$3^{...
- 328
3
votes
Accepted
Idea of base change for Division algebras over local field
An obvious idea to get a commutative square diagram, where $GL(n,F)$ and $GL(n,K)$ are respectively replaced by $D_F^\times$ and $D_K^\times$, is to use the Jacquet-Langlands transfer between ...
- 6,349
3
votes
Accepted
Representations of $SL_1(D),$ where $D$ a division algebra over a local field
An article by Shai Shechter recently appeared on Math ArXiv:
"Characters of the Norm-One Units of Local Division Algebras of Prime Degree"
https://arxiv.org/abs/1512.02448
- 6,349
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
Classification of finite-dimensional real super C*-algebras
I was looking for this a few years ago, and found it in the appendix of this paper.
El-kaïoum, M. Moutuou. "Graded Brauer groups of a groupoid with involution." Journal of Functional Analysis 266.5 (...
- 1,749
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
2
votes
Accepted
Charaterisation of quaternion algebras
Yes, this is a great and well-studied question with a super nice answer! See Theorem 3.5.1 of my book (http://quatalg.org). You can also say something in characteristic 2 (see Theorem 6.2.8) if you ...
- 3,009
2
votes
Accepted
Hermitian forms over $K\times K$
The situation is quite the same when $V$ is not free over $R$. You only need an extra parameter, namely the dimension $n_1$ of $(1, 0)V$ over $K$. The equivalence classes of Hermitian forms over $V$ ...
- 7,893
2
votes
A problem about extensions of division rings
I find that the answer of this problem is affirmative and obvious by a trick of tensors:
$D\otimes_{E} \bar{F} \cong (D\otimes_{F}F)\otimes_{E} \bar{F}\cong D\otimes_{F}(F\otimes_{E} \bar{F})\cong D\...
- 331
2
votes
Accepted
Regular elliptic elements are dense in p-adic division algebra
If $m_h$ is the minimal polynomial of $h$ over $F$, then $f_h$ is a power of $m_h$. (This fact is standard, but, to have a definite reference, I looked though my books on central simple algebras and ...
- 12.9k
1
vote
Properties of finite dimensional, real division algebras that yield only $\mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$ and $\mathbb{O}$
I spent a day looking into this myself and I didn't find anything.
Given a real unital composition algebra with a positive-definite form, Hurwitz's theorem will tell you that it must be one of those ...
- 111
1
vote
Accepted
The size of endomorphism rings and the relation to ordinariness of Abelian surfaces
The general reference for this sort of questions is Waterhouse, Abelian varieties over finite fields. Your question is answered in: Theorem 7.2. If $A$ is ordinary (and simple), then $\mathop{End}(A)$ ...
- 1,196
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