Example 1

Put A = \mathbf{Z} and L = \mathbf{Q}(p^{1/d}), where p is a prime number and d an integer \geqslant 2. Let B denote the integral closure of \mathbf{Z} in L. Since the polynomial T^d - p of \mathbf{Z}[T] is an Eisenstein polynomial for p\mathbf{Z}_{(p)}, there exists a unique prime ideal q of B above p\mathbf{Z}. There therefore exists a unique normalised discrete valuation v of the field \mathbf{Q}(p^{1/d}) such that v(p) > 0; one has [L : K] = v(p) = d, and B/q is a field with p elements. The ring B_q of the valuation v is equal to \mathbf{Z}_{(p)}[p^{1/d}].