2 Descent of principal ideals
For every number field \(L\), the ring \(\mathcal O_L\) is faithfully flat over \(\mathcal O_{L^+}\).
The proof uses the faithfully-flat criterion in terms of prime spectra. Flatness is available for the ring-of-integers extension. To prove faithfulness, one starts with a prime ideal of \(\mathcal O_{L^+}\) and applies lying-over to obtain a prime ideal of \(\mathcal O_L\) above it. The induced map on prime spectra is therefore surjective. This proves faithful flatness by the prime-spectrum criterion.
If \(J\) is an ideal of \(\mathcal O_{L^+}\), then extension to \(\mathcal O_L\) followed by contraction is the identity:
Apply the ideal-theoretic contraction lemma for faithful flatness, using theorem 2.1.
Let \(I\) be an ideal of \(\mathcal O_{K^+}\). If \(I\mathcal O_K\) is generated by the image of an element \(b_0\in \mathcal O_{K^+}\), then \(I\) is principal.
Contract the equality \(I\mathcal O_K=(b_0)\) from \(\mathcal O_K\) back to \(\mathcal O_{K^+}\). The left side contracts to \(I\) by theorem 2.2; the right side contracts to the ideal generated by \(b_0\). Hence \(I=(b_0)\).
If \(I\) is an ideal of \(\mathcal O_{K^+}\), then the extended ideal \(I\mathcal O_K\) is fixed by complex conjugation.
The extension is generated by elements coming from \(K^+\). Those elements are fixed by conjugation, so conjugating the extended ideal gives the same ideal.
Suppose \(I\mathcal O_K=(a)\) and, by conjugation stability, also \(I\mathcal O_K=(\overline a)\). Then \(a\) and \(\overline a\) are associated: there is a unit \(u\) with
Two nonzero generators of the same principal ideal in a domain differ by a unit. Applying this fact to the two displayed generators gives the result.
For the unit \(u\) obtained from \(\overline a=ua\), one has
Apply complex conjugation to \(\overline a=ua\) and use \(\overline{\overline a}=a\). Since \(a\ne 0\), cancellation gives \(u\overline u=1\).
Let \(K/\mathbb {Q}\) be the \(p\)th cyclotomic field with \(p\) odd. If \(u\in \mathcal O_K^\times \) satisfies \(u\overline u=1\), then
for some integers \(n,m\).
Such a unit has absolute value \(1\) at every archimedean embedding. Therefore it is a torsion unit. The torsion units in the prime cyclotomic field are generated by \(-1\) and the primitive \(p\)th root of unity.
Under the hypotheses above, the generator \(a\) may be replaced by an associate \(b\) such that
The quotient \(\overline a/a\) is the antisymmetric unit \(u\). By theorem 2.7, \(u\) is a sign times a power of \(\zeta _p\). Multiplying \(a\) by a suitable power of \(\zeta _p\) changes the generator by a unit and kills the conjugation defect.
If \(b\in \mathcal O_K\) is fixed by complex conjugation, then it is the image of an element of \(\mathcal O_{K^+}\).
Being fixed by conjugation is the defining condition for membership in the maximal real subfield. Since \(b\) is already integral over \(\mathbb {Z}\), the descended element lies in the ring of integers of \(K^+\).
Let \(K\) be the \(p\)th cyclotomic field, with \(p\) odd. If an ideal of \(\mathcal O_{K^+}\) becomes principal after extension to \(\mathcal O_K\), then it was already principal.
Choose a generator \(a\) of the extended ideal. Conjugation stability gives the same ideal generated by \(\overline a\), so theorem 2.5 produces the unit ratio \(u\). The relation theorem 2.6 and the cyclotomic classification theorem 2.7 allow one to replace \(a\) by a conjugation-fixed associate \(b\). By theorem 2.9, \(b\) descends to an element of \(\mathcal O_{K^+}\). Finally theorem 2.3 contracts the generated ideal back to \(\mathcal O_{K^+}\).