# Questions tagged [quaternion-algebras]

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### Proof that the left ideal I of prime norm in maximal order can be written as I = ON + Oα

It seems there is a well-known fact that if $O$ is a maximal order in quaternion algebra $B$ and $I$ is a left $O-$ideal such that $nrd(I) = N$ is prime, then $I = ON + Oα$ with $gcd(N^2, nrd(α)) = N$....
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### Group of invertible elements in a degree 4 central simple algebra with symplectic involution with norm in a center

Let $A$ be a central simple algebra of degree 4 (i.e. dimension 16) over a field $F$ with $\mathrm{char}(F) \neq 2$. It is known that any such algebra is a tensor product $D_1 \otimes D_2$ of two ...
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### A note on orders in quaternion algebras

Definition. An algebra $B$ over a field $F$ is a quaternion algebra if there exists $i,j\in B$ such that $1,i,j,ij$ is a basis for $B$ as a vector space over $F$, where $i^2=a,j^2=b;a,b\in F^\times$. ...
• 175
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### Maximal orders and surface subgroups of even genus

Let $A$ be a quaternion algebra over a totally real number field $k$. Suppose that $A$ splits at exactly one real place of $A$. Let $\mathcal{O}$ be a maximal order in $A$. Then $\mathcal{O}$ contains ...
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### Motivating unpublished statements of Gauss about congruences and quaternions

Background Modern claims that Gauss anticipated the quaternions algebra are based primarily on an unpublished fragment of Gauss dated to 1819 and entitled "rotations of space". In this ...
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### Abelianizations of arithmetic Fuchsian groups

Let $a,b$ be positive integers with $x^2-ay^2-bz^2+abv^2=0$ having only the zero solution over $\mathbb Z$ and consider the Fuchsian group \begin{equation*} \Gamma=\left\{\begin{bmatrix} k+\sqrt{a}l &...
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### Number of cosets by quaternion order

Let $D$ be a quaternion division algebra over $\mathbb Q$ and let $\Lambda\subset D$ be a maximal order. For $n\in\mathbb Z$ let $\Lambda^{(n)}$ denote the set of all $\lambda\in\Lambda$ with reduced ...
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1 vote
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### Is the isomorphism class of Eichler order with prime level in $B_{p,\infty}$ with extra units at most unique?

Let $B_{p, \infty}$ be the quaternion algebra over $\mathbb{Q}$ ramified exactly at $p$ and $\infty$. Let $\mathcal{O}$ be an order in $B_{p, \infty}$. As we know, if $\mathcal{O}$ is maximal, then ...
555 views

### How to prove that a quaternion algebra over ℤₚ is isomorphic to Mat₂(ℤₚ) for p prime?

How to prove without using advanced theorems that quaternions algebra $H = \genfrac(){}{}{-1,-1}{\mathbb{Z}_p}$, where $p$ is prime that $H \cong\operatorname{Mat}_2({\mathbb{Z}_p})$? My ideas: I ...
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### Are there any central simple algebras admitting a standard basis?

Are there any central simple algebras admitting a standard basis? By a standard basis I mean a normal basis that has a cyclic property generalizing that of the familiar basis $1, i, j, k$ for ...
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### Ramification of quaternion algebras over $\mathbb Q$

I'm reading on the classification of quaternion algebras over $\mathbb{Q}$. In the most common definition, we have a quaternion algebra $Q = \left(\frac{a,b}{\mathbb{Q}} \right)$ splits at a finite ...
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### Bézout theorem for complex solutions of systems of homogeneous quaternionic polynomials

Suppose that I have a system of $n$ homogeneous polynomials in $n+1$ variables $p_{1},\dots,p_{n}\in\mathbb{H}[x_{0},\dots,x_{n}]$ with coefficients at both sides in $\mathbb{H}.$ I want to know if ...
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