\(\mathcal O_L\) is a finitely generated torsion-free module over the Dedekind domain \(\mathcal O_{L^+}\), hence projective, hence flat. For faithfulness we use the criterion in terms of prime spectra: it suffices that \(\operatorname {Spec}\mathcal O_L \to \operatorname {Spec}\mathcal O_{L^+}\) be surjective. Given a prime \(\mathfrak p\) of \(\mathcal O_{L^+}\), the extension \(\mathcal O_{L^+} \subseteq \mathcal O_L\) is integral, so lying-over produces a prime of \(\mathcal O_L\) contracting to \(\mathfrak p\).

For a faithfully flat ring map \(A \to B\), every ideal \(J \subseteq A\) satisfies \(JB \cap A = J\): the inclusion \(J \subseteq JB \cap A\) is clear, and the quotient map \(A/J \to B/JB\) is injective by faithful flatness of \(B/JB\) over \(A/J\). Apply this with theorem 2.1.

Contract the equality \(I\mathcal O_K=(b_0)\) back to \(\mathcal O_{K^+}\). The left-hand side contracts to \(I\) by theorem 2.2. For the right-hand side, the contraction of \(b_0\mathcal O_K\) contains \(b_0\mathcal O_{K^+}\), and conversely any element of \(b_0 \mathcal O_K \cap \mathcal O_{K^+}\) is of the form \(b_0 c\) with \(c \in K \cap \mathcal O_K = \mathcal O_{K^+}\) (using \(b_0 \ne 0\); if \(b_0 = 0\) then \(I = 0\) is principal anyway). Hence \(I=(b_0)\).

The extension \(I\mathcal O_K\) is generated by elements coming from \(K^+\), and those elements are fixed by conjugation. Conjugation is a ring automorphism of \(\mathcal O_K\), so it maps \(I\mathcal O_K\) onto the ideal generated by the conjugates of the generators, which is \(I\mathcal O_K\) itself.

Two generators of the same nonzero principal ideal in a domain differ by a unit: from \((a) = (\overline a)\) we get \(\overline a = ua\) and \(a = v \overline a\) for some \(u, v \in \mathcal O_K\), whence \(a = vua\) and, since \(a \ne 0\), \(vu = 1\).

Apply complex conjugation to \(\overline a=ua\): since conjugation is an involution, \(a = \overline u\, \overline a = \overline u u a\). As \(a\ne 0\), cancellation in the domain \(\mathcal O_K\) gives \(u\overline u=1\).

At every complex embedding \(\sigma \) of \(K\), conjugation corresponds to complex conjugation of the image, so \(|\sigma (u)|^2 = \sigma (u \overline u) = 1\). An algebraic integer all of whose conjugates have absolute value \(1\) is a root of unity (Kronecker’s theorem), so \(u\) is a torsion unit. The torsion subgroup of \(\mathbb {Q}(\zeta _p)^\times \) for \(p\) odd is generated by \(-1\) and \(\zeta _p\), which gives the displayed form.

By theorem 2.7, the unit \(u = \overline a / a\) has the form \((-1)^n \zeta _p^m\). Since \(p\) is odd, every power of \(\zeta _p\) is a square in \(\langle \zeta _p\rangle \), and the sign can be absorbed: write \(u = w/\overline w\) for a suitable root of unity \(w\) (concretely, a power of \(\zeta _p\) chosen so that \(w^2\) accounts for \(\zeta _p^m\), possibly times \(-1\)). Then \(b := w a\) satisfies \(\overline b = \overline w\, \overline a = \overline w u a = w a = b\), and \(b\) is an associate of \(a\), so it generates the same ideal.

For a CM field, \(K^+\) is exactly the fixed field of conjugation, so the element \(b\) lies in \(K^+\). Being integral over \(\mathbb {Z}\), it lies in \(K^+ \cap \mathcal O_K = \mathcal O_{K^+}\).

Let \(I\) be such an ideal; we may assume \(I \ne 0\). Choose a generator \(a\) of \(I\mathcal O_K\). The extended ideal is conjugation-stable (theorem 2.4), so theorem 2.5 produces a unit \(u\) with \(\overline a = ua\), and theorem 2.6 shows \(u\overline u = 1\). The classification of such units (theorem 2.7) lets us replace \(a\) by a conjugation-fixed associate \(b\) with \((b) = I\mathcal O_K\) (theorem 2.8). By theorem 2.9, \(b\) descends to an element \(b_0 \in \mathcal O_{K^+}\), and theorem 2.3 contracts the equality \(I\mathcal O_K = (b_0)\) to \(I = (b_0)\).