Corollary 2

— A field that is finitely generated over its prime subfield contains only finitely many roots of unity.

By Cor. 1 we are reduced to proving that a field L which is an extension of finite degree of its prime subfield possesses only a finite number of roots of unity. This is clear when L is of characteristic $\neq 0$ because it is then finite. If L is of characteristic 0 and contains infinitely many roots of unity, then it contains primitive roots of unity of arbitrarily high order. By V, p. 84, Th. 2 there exist therefore an infinity of integers $n > 0$ such that $\varphi(n) \leq [L : \mathbf{Q}]$, which is absurd (V, p. 80, Formulae (2) and (3)).