The Hamilton quaternions are a quaternion algebra ( in the above sense ) over F = \ mathbb { R } ( the real number field ), and indeed the only one over \ mathbb { R } apart from the 2? real matrix algebra, up to isomorphism.
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The first order zero conjectures are used in recent versions of the PARI / GP computer algebra system to compute Hilbert class fields of totally real number fields, and the conjectures provide one solution to Hilbert's twelfth problem, which challenged mathematicians to show how class fields may be constructed over any number field by the methods of complex analysis.
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