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On linear independence of trigonometric numbers. (English) Zbl 1449.11078

Summary: A necessary and sufficient condition is established for 1, cos\((\pi r_1)\), and cos\((\pi r_2)\) to be rationally independent, where \(r_1, r_2\) are rational numbers. The elementary computational argument yields linear independence over larger number fields as well.

MSC:

11J72 Irrationality; linear independence over a field
11R11 Quadratic extensions
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