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The aim of this paper is to show that the simplest techniques of linear algebra allow us to make explicit the defining equations of the maximal real cyclotomic extensions ℚ(ζ + ζ−1 of ℚ(ζ), where ζ stands for a primitivep ν-th rooot of unity withp a rational prime and ν any positive integer.

On the defining polynomials of maximal real cyclotomic extensions