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Questions tagged [division-algebras]

A division algebra $D$ is a vector spaces over a field $F$ equipped with a bilinear product and a multiplicative neutral element $1$. All the non-zero elements of $D$ have a multiplicative inverse. Associativity is often assumed but not always. Any field is a commutative, associative division algebra. A skewfield = a division ring is always a division algebra over its center. The quaternions form the best known non-commutative division algebra.

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Extending valuation to division algebra over a non-Archimedean local field?

Let $D$ be a division algebra over a non-Archimedean local field $K$. I would like to extend the discrete valuation on $K$ to $D$. For any $x \in D$, the subfield $K(x)$ of $D$ has a unique ...
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Is every endomorphism of the quaternion ring surjective?

Is the quaternion ring an EAS Division ring? An EAS Division ring is a ring $D$ such that each endomorphism of $D$ is surjective. I know that $\mathbb{R}$ and $\mathbb{Q}$ are EAS Division rings.
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reduced norm is proper

If $D$ is a central division algebra of dimension $n^2$ over a field $k$ we can consider the reduced norm $\nu : D \to k$, which satisfies $\nu^n = N_{D/k}$. In particular we get a group homomorphism $...
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An example of a division ring $D$ that is **not** isomorphic to its opposite ring

I recall reading in an abstract algebra text two years ago (when I had the pleasure to learn this beautiful subject) that there exists a division ring $D$ that is not isomorphic to its opposite ring. ...
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linear algebra over a division ring vs. over a field

When I was studying linear algebra in the first year, from what I remember, vector spaces were always defined over a field, which was in every single concrete example equal to either $\mathbb{R}$ or $\...
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What are some real-world uses of Octonions?

... octonions are the crazy old uncle nobody lets out of the attic: they are nonassociative. Comes from a a quote by John Baez. Clearly, the sucessor to quaterions from the Cayley-Dickson process is ...