# Tagged Questions

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### Construction of Hyper-Complex Numbers

How does one construct a hyper-complex number multiplication table? For example: Quarternions: ...
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### Tensor products and isomorphic algebras

I found that $\mathbb{C} \otimes_{\mathbb{R}} \mathbb{C} \simeq \mathbb{C} \oplus\mathbb{C}$ and that $\mathbb{H} \otimes_{ \mathbb{R}} \mathbb{C} \simeq M_2( \mathbb{C})$. Could anybody hint me how ...
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### Need help with this exercise about real division algebra

I am trying to solve the following exrcise: Prove that complex multiplication does not extend to a multiplication on $\mathbb R^3$ so as to make $\mathbb R^3$ into a real division algebra. I ...
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### Isomorphism of quaternions with a matrix ring over real numbers

Let $\mathcal A$ be the algebra over the real numbers consisting of matrices of the form $$\begin{pmatrix} z&w\\ - \bar{w}& \bar{z} \end{pmatrix} \ (z, w \in \mathbb C).$$ $\mathcal A$ is in ...
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### Is there a (hypercomplex) number system, in which addition is **not** commutative

Now, we all know and love the quaternions, in which multiplication is not commutative, and the fact that, in all Clifford algebras, exponentiation is not commutative. Having looked at the properties ...
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### Proof that the subalgebra generated by any two elements of $\mathbb{O}$ is isomorphic to $\mathbb{R}$, $\mathbb{C}$, or $\mathbb{H}$

In the wikipedia atrticle (http://en.wikipedia.org/wiki/Octonion) it is stated that "one can show that the subalgebra generated by any two elements of $\mathbb{O}$ is isomorphic to $\mathbb{R}$, ...
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### Prove that $Q$ is a group under quaternion multiplication

Consider the subset $Q$ of the quaternions defined by $$Q=\{1,-1,i,-i,j,-j,k,-k\}.$$ Show that $Q$ is a group under quaternion multiplication. I know to prove something's a group, you must show ...
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### Analogy between the quaternion ring and extensions of the rationals

I've started studing fields and their extensions. As an exercise I proved that $[\mathbb{Q}(\sqrt2,\sqrt3):\mathbb{Q}]=4$ by showing that $B=(1,\sqrt2,\sqrt3,\sqrt6)$ is a base for the extension field ...
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### Show that $Q_8$ can't be embedded in $M_{2 \times 2}(\mathbb{R})$ as a group.

So, suppose that we're working in a field $F$. Consider the ring $M_{n \times n} (F)$ which is the set all $n \times n$ matrices with entries in $F$. Is it possible to determine whether a matrix ...
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### non-division quaternion algebra is isomorphic to $2\times 2$ matrices

Let $k$ be a field of characteristic $\neq2$. Let $a,b\in k$ be nonzero elements. Let $A:=\left(\frac{a,b}{k}\right)$ be the quaternion algebra over $k$ with parameters $a,b$. Suppose $A$ is not a ...
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### Indefinite quaternion algebra over Q

Let $D$ be an indefinite quaternion algebra over $\Bbb Q$. We have a chosen isomorphism $\iota \colon D \otimes_{\Bbb Q} {\Bbb R} \cong {\mathrm{M}}_2({\Bbb R})$. Q: If we choose another isomorphism ...
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### Unit quaternions as rotations

How would one represent the map $f$ such that $f(1) = i, f(i) = -1$ and keeping $j$ and $k$ fixed as a quaternion representation of rotations?
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### What specific algebraic properties are broken at each Cayley-Dickson stage beyond octonions?

I'm starting to come around to an understanding of hypercomplex numbers, and I'm particularly fascinated by the fact that certain algebraic properties are broken as we move through each of the $2^n$ ...
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### Quaternions as an Algebra

I'm lacking some vital understanding about quaternions and algebras in general. If we first define $V=\{a+bi+cj+dk|a,b,c,d\in\mathbb{R}\}$. Then we define scalar multiplication, vector multiplication, ...
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### Every element with finite conjugates in the ring of real quaternions is a real number

Let $H$ be the ring of real quaternions and let $x$ be a member of $H$ having finite conjugates. Prove that $x$ is a real number. I worked a lot on this question, but no progress! :|
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### Show that for $\forall a\in\mathbb{H}, \ \exists b \in\mathbb{H}: ab =ba = 1$.

Show that $\forall a\in\mathbb{H}, \ \exists b \in\mathbb{H}: ab =ba = 1.$ I am pretty sure I can easily google the multiplicative inverse in $\mathbb{H}$, but can you give me a hint on how to ...
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### Enumeration of Hurwitz quaternions of norm p

In "on Quaternions and Octonions" by Conway and Smith, they quote a result by which for each prime norm $p$ there are exactly $p+1$ Hurwitz quaternions of norm $p$. I haven't found any proof of that. ...
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### Examples of a non commutative division ring

What are some examples of a non commutative division ring other than quaternions?
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### Automorphism of $Q_8$ [duplicate]

Is there anyone could help me to prove that $Aut(Q_8)=S_4$? Someone told me that there's an isomorphism between the rigid motions of cube and $Aut(Q_8)$, any ideas? Thank you!
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### What lies beyond the Sedenions

In the construction of types of numbers, we have the following sequence: $$\mathbb{R} \subset \mathbb{C} \subset \mathbb{H} \subset \mathbb{O} \subset \mathbb{S}$$ or: 2^0 \mathrm{-ions} \subset ...
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### what is the tensor product $\mathbb{H\otimes_{R}H}$

I'm looking for a simpler way of thinking about the tensor product: $\mathbb{H\otimes_{R}H}$, i.e a more known algbera which is isomorphic to it. I have built the algebra and played with it for a ...
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### Why is quaternion algebra 4d and not 3d?

Why is quaternion algebra 4D and not 3D? Complex algebra is 2D and what is known as quaternion algebra jumps to 4D. $i^2 = j^2 = k^2 = ijk = -1$ Using $1, i, j,$ and $k$ as the base (where ...