The quaternions tag has no usage guidance.

**9**

votes

**2**answers

1k views

### When the determinant of a 2x2 polynomial matrix is a square?

Consider a 2x2 matrix $A$ with entries from $\mathbb{C}[x,y]$. Assume that $\mathrm{det} A$ is a square. Is it true that then $A$ can be represented as a noncomuting product $A=A_1 A_2 … A_{2n}$, in ...

**19**

votes

**1**answer

1k views

### The Quaternion Moat Problem

"One cannot walk to infinity on the real line if one uses steps of bounded
length and steps on the prime numbers. This is simply
a restatement of the classic result that there are arbitrarily
large ...

**2**

votes

**1**answer

604 views

### Central division and quaternion algebras

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 Schur index and $exp$ ...

**3**

votes

**3**answers

603 views

### Polynomial Vector Fields on the 3-Sphere

EDIT(3): I am looking for a basis for the Lie algebra of polynomial vector fields on $S^3$.
EDIT(2): I am fairly certain now that my question is more along the lines of, what does the Lie algebra of ...

**0**

votes

**1**answer

908 views

### Quaternion between two quaternions [closed]

Hello,
I have an orientation P1 in a 3D space, represented as a quaternion [w x y z].
Then P1 is rotated using another quaternion (q1) with the formula
...

**2**

votes

**1**answer

2k views

### In the quaternions, “any imaginary unit may be called i”

Introduction
Suppose we are trying to prove that $\rm PSO_3\times PSO_3$ is isomorphic with $\rm PSO_4,$ and we catch on to the idea of using the quaternions to do so. We realize (as in Conway & ...

**35**

votes

**5**answers

2k views

### Is there a quaternionic algebraic geometry ?

Let $\mathbb{H}$ be the skew-field of quaternions. I'm aware of the
Theorem 1. A function $f:\mathbb{H}\to\mathbb{H}$ which is $\mathbb{H}$-differentiable on the left (i.e. the usual limit ...

**16**

votes

**1**answer

1k views

### William Rowan Hamilton and Algebra as Time

This question ended up longer than I intended (though most of the bulk is interesting remarks by Hamilton), so I thought it might be good to include my question at the beginning before the ...

**5**

votes

**5**answers

1k views

### Why is every quadratic subfield of a Galois extension of the rationals with the quaternions as Galois group real?

Suppose that L is a field extension of the rationals with Galois group the quaternions Q={1,-1,i,-i,j,-j,k,-k}. Furthermore assume that L contains a quadratic subfield K. I have learned from this link ...

**11**

votes

**2**answers

2k views

### Gaussian primes, quaternion primes, … octonions?

Is there a notion of an octonion prime?
A Gaussian integer $a + bi$ with $a$ and $b$ nonzero is prime if $a^2 + b^2$ is prime.
A quaternion $a + bi + cj + dk$ is prime iff $a^2 + b^2 +c^2 + d^2$ is ...

**10**

votes

**1**answer

437 views

### What rings/groups have interesting quaternionic representations?

Let $\mathbb{H}$ denote the quaternions. Let $G$ be a group, and define a representation of $G$ on $\mathbb{H}^n$ in the natural way; that is, its a map $\rho:G\rightarrow ...