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Let <math>B</math> be a quaternion algebra over <math>\mathbb{Q}</math> and let <math>\nu</math> be a [[Place (mathematics)|place]] of <math>\mathbb{Q}</math>, with completion <math>\mathbb{Q}_\nu</math> (so it is either the ''p''-adic numbers <math>\mathbb{Q}_p</math> for some prime ''p'' or the real numbers <math>\mathbb{R}</math>). Define <math>B_\nu:= \mathbb{Q}_\nu \otimes_{\mathbb{Q}} B</math>, which is a quaternion algebra over <math>\mathbb{Q}_\nu</math>. So there are two choices for
<math>B_\nu</math>: the 2 by 2 matrices over <math>\mathbb{Q}_\nu</math> or a [[division algebra]].
We say that <math>B</math> is '''split''' (or '''unramified''') at <math>\nu</math> if <math>B_\nu</math> is isomorphic to the 2×2 matrices over <math>\mathbb{Q}_\nu</math>. We say that ''B'' is '''non-split''' (or '''ramified''') at <math>\nu</math> if <math>B_\nu</math> is the quaternion division algebra over <math>\mathbb{Q}_\nu</math>. For example, the rational Hamilton quaternions is non-split at 2 and at <math>\infty</math> and split at all odd primes.
A quaternion algebra over the rationals which splits at <math>\infty</math> is analogous to a real [[quadratic field]] and one which is non-split at <math>\infty</math> is analogous to an imaginary quadratic field.
it is infinite if the quaternion algebra splits at <math>\infty</math>{{Citation needed|date=July 2009}} and it is finite otherwise{{Citation needed|date=July 2009}}, just as the unit group of an order in a quadratic ring is infinite in the real quadratic case and finite otherwise.
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