The reduced norm is multiplicative and the reduced trace is additive.
2.
Map " A " to a matrix ring over a splitting field and define the reduced norm and trace to be the composite of this map with determinant and trace respectively.
3.
Let \ mathcal O ^ 1 be the group of elements in \ mathcal O of reduced norm 1 and let \ Gamma be its image in M _ 2 ( \ mathbb C ) via \ phi.
4.
Let \ mathcal O ^ 1 be the group of elements in \ mathcal O of reduced norm 1 and let \ Gamma be its image in M _ 2 ( \ mathbb R ) via \ phi.
5.
An element " a " of " A " is invertible if and only if its reduced norm in non-zero : hence a CSA is a division algebra if and only if the reduced norm is non-zero on the non-zero elements.
6.
An element " a " of " A " is invertible if and only if its reduced norm in non-zero : hence a CSA is a division algebra if and only if the reduced norm is non-zero on the non-zero elements.
7.
Then the image of \ Gamma is a subgroup of \ mathrm { SL } _ 2 ( \ mathbb R ) ( since the reduced norm of a matrix algebra is just the determinant ) and we can consider the Fuchsian group which is its image in \ mathrm { PSL } _ 2 ( \ mathbb R ).
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