Suppose that $Q$ is a quaternion division algebra with center $k$, where $k$ is an arbitrary commutative field (let's say with $\operatorname{char}(k) \neq 2$ if necessary). Assume …

This has been bugging me for a while. According to https://en.wikipedia.org/wiki/Euclidean_division, if I divide integer $a$ by integer $b$, I get unique $t$, $r$ such that $a = t …

Let D be a division algebra over F, E its maximal field. Is it true that: 1) all such fields are equivalent over F? 2) all such fields are conjugate by inner automorphisms of D?

A celebrated theorem of Milnor and Kervaire asserts that any finite dimensional division algebra over the real numbers has dimension 1,2,4 or 8. This result is established using me …

I would like to know if there are some central-simple algebras $D_1$, $D_2$ and $D_3$ over a field $k$ satisfying the following properties : $ind(D_1)=exp(D_1)=4$ ($ind$ is the S …

Suppose $D$ is an explicitly given rank 9 central simple algebra over ${\mathbb Q}$ (or a number field). For example it could be specified by two cubic subfields ${\mathbb Q}(a), { …

The product of a quaternion multiplied by a real number is a quaternion, but the product of a quaternion multiplied by a complex number is not in general a quaternion. Why are the …

Let's say a normed division algebra is a real vector space $A$ equipped with a bilinear product, an element $1$ such that $1a = a = a1$, and a norm obeying $|ab| = |a| |b|$. The …

If $F$ is a global field then a standard exact sequence relating the Brauer groups of $F$ and its completions is the following: $$0\to Br(F)\to\oplus_v Br(F_v)\to\mathbf{Q}/\mathb …

Let $K$ be a skew-field, infinite dimensional over its center $F$. From Kaplansky's PI-theorem it then follows that $K$ cannot satisfy a polynomial identity (the theorem says tha …

For any matrices $A$,$B$ in $SL(2,\mathbb{C})$, let $M(A,B)$ be the 4 by 4 matrix whose columns are matrices $I$,$A$,$B$,$AB$. Then it is not hard to verify that $det M(A,B)= 2- tr …

Let $k$ be a field, $K/k$ a separable quadratic extension, and $D/K$ a central division algebra of dimension $r^2$ over $K$ with an involution $\sigma$ of second kind (i.e. $\sigma …

This is really an irrelevant question in the sense that the answer isn't remotely "logically crucial for the Langlands programme" or whatever---it's just something that occurred to …

D is a central division algebra over F. We know that we can always find a maximal subfield K inside D such that K/F is separable. I want to know can we always make it Galois?