A functional equation related to the product in a quadratic number field |
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| Authors: | Lucio R Berrone Luis V Dieulefait |
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| Institution: | 1. Laboratorio de Ac??stica y Electroac??stica, Facultad de Ciencias Exactas, Ing. y Agrim., Consejo Nacional de Investigaciones Cient??ficas y T??cnicas (CONICET), Universidad Nacional de Rosario, Riobamba 245 bis, 2000, Rosario, Argentina
2. Departament d??Algebra i Geometria, Universitat de Barcelona, G. V. de les Corts Catalanes 585, 08007, Barcelona, Spain
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| Abstract: | The functional equation $$f(x_{1},y_{1})f(x_{2},y_{2})=f(x_{1}x_{2}+\alpha y_{1}y_{2},x_{1}y_{2}+x_{2}y_{1}),\ (x_{1},y_{1}),\,(x_{2},y_{2})\in \mathbb{ R}^{2}$$ arises from the formula for the product of two numbers in the quadratic field ${\mathbb{Q}(\sqrt{\alpha})}$ . The general solution ${f:\mathbb{R}\rightarrow \mathbb{R}}$ to this equation is determined. Moreover, it is shown that no more general equations arise from a change of basis in the field. |
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