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LSpice
LSpice
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How to prove that a quaternion algebra over Zpℤₚ is isomorphic to Mat2Mat₂(Zpℤₚ) for p prime?

How to prove without using of advanced theorems that quaternions algebra $H = \left(\frac{-1,-1}{\mathbb{Z}_p} \right)$$H = \genfrac(){}{}{-1,-1}{\mathbb{Z}_p}$, where $p$ is prime that H $\cong\mathrm{Mat}_2({\mathbb{Z}_p})$$H \cong\operatorname{Mat}_2({\mathbb{Z}_p})$?

My ideas: I tried to build an explicit isomorphism, but as I think it is only possible when $p = 1$ (mod 4)$p = 1 \pmod 4$, and for $p = 1$ (mod 4)$p = 1 \pmod 4$ it get it. In my second attempt, I tried to look at them as vector spaces of the same dimension.

How to prove that quaternion algebra over Zp isomorphic to Mat2(Zp) for p prime

How to prove without using of advanced theorems that quaternions algebra $H = \left(\frac{-1,-1}{\mathbb{Z}_p} \right)$, where $p$ is prime that H $\cong\mathrm{Mat}_2({\mathbb{Z}_p})$

My ideas: I tried to build an explicit isomorphism, but as I think it is only possible when $p = 1$ (mod 4), and for $p = 1$ (mod 4) it get it. In my second attempt, I tried to look at them as vector spaces of the same dimension.

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 tried to build an explicit isomorphism, but as I think it is only possible when $p = 1 \pmod 4$, and for $p = 1 \pmod 4$ it get it. In my second attempt, I tried to look at them as vector spaces of the same dimension.

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YCor
YCor
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How to prove that Quaternion'squaternion algebra over Zp isomorphic to Mat2(Zp), for p - prime

How to prove without using of advanced theorems that quaternions algebra $H = \left(\frac{-1,-1}{\mathbb{Zp}} \right)$$H = \left(\frac{-1,-1}{\mathbb{Z}_p} \right)$,where p where -$p$ is prime that H $\cong$ $Mat_2({\mathbb{Zp}})$$\cong\mathrm{Mat}_2({\mathbb{Z}_p})$

My ideas: I tried to build an explicit isomorphism, but as I think it is only possible when p = 1$p = 1$ (mod 4), and for p = 1$p = 1$ (mod 4) it get it. In my second attempt, I tried to look at them as vector spaces of the same dimension.

How to prove that Quaternion's algebra over Zp isomorphic to Mat2(Zp), p - prime

How to prove without using of advanced theorems that quaternions algebra $H = \left(\frac{-1,-1}{\mathbb{Zp}} \right)$,where p - is prime that H $\cong$ $Mat_2({\mathbb{Zp}})$

My ideas: I tried to build an explicit isomorphism, but as I think it is only possible when p = 1 (mod 4), and for p = 1 (mod 4) it get it. In my second attempt, I tried to look at them as vector spaces of the same dimension.

How to prove that quaternion algebra over Zp isomorphic to Mat2(Zp) for p prime

How to prove without using of advanced theorems that quaternions algebra $H = \left(\frac{-1,-1}{\mathbb{Z}_p} \right)$, where $p$ is prime that H $\cong\mathrm{Mat}_2({\mathbb{Z}_p})$

My ideas: I tried to build an explicit isomorphism, but as I think it is only possible when $p = 1$ (mod 4), and for $p = 1$ (mod 4) it get it. In my second attempt, I tried to look at them as vector spaces of the same dimension.

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Joseph Jordan
Joseph Jordan
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How to prove that Quaternion's algebra over Zp isomorphic to Mat2(Zp), p - prime

How to prove without using of advanced theorems that quaternions algebra $H = \left(\frac{-1,-1}{\mathbb{Zp}} \right)$,where p - is prime that H $\cong$ $Mat_2({\mathbb{Zp}})$

My ideas: I tried to build an explicit isomorphism, but as I think it is only possible when p = 1 (mod 4), and for p = 1 (mod 4) it get it. In my second attempt, I tried to look at them as vector spaces of the same dimension.