2 Exact \(p\)-saturation
The ambient plus-side unit subgroup used for saturation is the full group \((\mathcal O_{K^+})^\times \).
For a subgroup \(H\) of an abelian group, \(H^p\) denotes the exact image of the \(p\)th-power map:
A subgroup \(H\le E\) is \(p\)-saturated in \(E\) when
Equivalently, every element of \(H\) that is a \(p\)th power in \(E\) is already a \(p\)th power inside \(H\).
An element written in the standard generators has the form
This expression is packaged with integer exponents \(s,e_a\).
Every element of \(C^+\) admits such an integer-exponent expression.
This is induction on membership in the subgroup generated by \(-1\) and the finite family of real cyclotomic units. The proof checks the generators, closure under multiplication, inverses, and the identity element.
Assume that whenever
is a \(p\)th power in the full unit group, every exponent \(e_a\) is zero modulo \(p\). Then \(C^+\) is \(p\)-saturated in the full plus-side unit group.
Take an element of \(C^+\) which is a \(p\)th power in the ambient unit group. Write it as an exponent product using theorem 2.5. The hypothesis says each \(e_a\) is divisible by \(p\). Divide the exponents by \(p\) to construct an element of \(C^+\) whose \(p\)th power is the original element.
If \(C^+\) is \(p\)-saturated in the full plus-side unit group, then
The group-theoretic input says that a finite-index \(p\)-saturated subgroup has index prime to \(p\) when all ambient \(p\)-torsion already lies in the subgroup. For \(C^+\), the torsion condition follows from the description of torsion units in \(K^+\), and finite index is supplied by the cyclotomic-unit index setup.