Rang de l'image du groupe des unités et polynômes lacunaires

Amorosointroduced a new class of number fields, the so-called class of $ PCM $-fields (proche d'un corps $CM$), having interesting arithmetic proprierties. The main result of our work is the proof of the fact that no totally real field is $PCM$. Moreover we show how our proof, in the special case of finite totally real abelian extensions of $mathbb{Q}$, implies a lower bound for the number of non-zero coefficients of polynomials whose certain zeros are roots of $1$.