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Let BB be a quaternion algebra over Q\Q, and let OBO\subset B be an order. Let x1,x2,x3,x4Ox_1,x_2,x_3,x_4 \in O be a Z\Z-basis of OO. The discriminant of OO is the integer disc(O)det(trd(xixj))i,j. \operatorname{disc}(O) \colonequals \left|\det(\operatorname{trd}(x_ix_j))_{i,j}\right|. where trd:BQ\operatorname{trd} : B \to \Q is the reduced trace.

The reduced discriminant of OO is the positive integer discrd(O)\operatorname{discrd}(O) such that discrd(O)2=disc(O). \operatorname{discrd}(O)^2 = \operatorname{disc}(O).

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  • Last edited by Jacob Swenberg on 2024-02-09 13:26:37
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