Proposition 11

— Suppose that K is infinite and let P : N → K be a polynomial mapping which is non-zero on K. There exists x ∈ N such that P(x) ≠ 0 and {x} is a basis of N over $\mathbf{K}^{(\Gamma)}$.

B) Suppose that K is finite. By Prop. 4 (V, p. 95)$^1$ every extension of finite degree over K has a cyclic Galois group. We shall therefore more generally consider the case where the group $\Gamma$ is cyclic of order n; we denote by $\gamma$ a generator of $\Gamma$.

The following lemma is a particular case of more general results proved in Chapter VII. The ring $A$ is either the ring $\mathbf{Z}$ of rational integers or the ring $K[X]$ of polynomials over the field $K$.