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We classify all Kutasov-Seiberg type dualities in large $N_c$ SQCD withadjoints of rational $R$-charges. This is done by equating the superconformalindex of the electric and magnetic theories: the obtained equation has asolution each time some product of cyclotomic polynomials has only positivecoefficients. In this way we easily reproduce without any reference to thesuperpotential or the choice of the equations of motion (classical chiral ring)all the known dualities from the literature, while adding to them a new familywith two adjoints with $R$ charges $\frac{2}{2k+1}$ and $\frac{2(k+1)}{2k+1}$for all integers $k>1$. We argue that these new fixed points could be in theirappropriate conformal windows and in some range of the Yukawas involved a lowenergy limit of the $D_{2k+2}$ fixed point. We try to clarify some issuesconnected to the difference between classical and quantum chiral ring of thisnew solution.

Kutasov-Seiberg dualities and cyclotomic polynomials