The totally real number fields play a significant special role in algebraic number theory.
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Let F be a totally real number field and A a quaternion algebra over F satisfying the following conditions.
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This was proved by for "'Q "', and for all totally real number fields by.
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In applications such as computer vision where the underlying field is the real number field, " homography " and " collineation " can be used interchangeably.
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On the other hand there are still some arithmetic constraints on the Veech group of a Veech surface : for example its trace field must be a totally real number field.
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An arithmetic Fuchsian group is constructed from the following data : a totally real number field F, a quaternion algebra A over F and an order \ mathcal O in A.
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Since the real number field is not algebraically closed, the geometry of even a plane curve " C " in the real projective plane is not a very easy topic.
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These days these are associated to algebraic groups ( respectively the Weil restriction from a totally real number field of, and the symplectic group ), for which it happens that automorphic representations can be derived from analytic functions.
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Since there are no non-trivial field automorphisms of the real number field, all the collineations are homographies in the real projective plane ., however due to the field automorphism complex conjugation, not all collineations of the complex projective line are homographies.
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"' Hilbert modular forms "'are functions in " n " variables, each a complex number in the upper half-plane, satisfying a modular relation for 2 & times; 2 matrices with entries in a totally real number field.
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