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The divisibility is between generators of characteristic ideals. Auxiliary-prime derivative operators applied to a root-of-unity Euler system produce Kummer classes with controlled valuations. The congruence axiom identifies the valuation at a new auxiliary prime with a residue symbol of the preceding class. The Chebotarev density theorem chooses prime representatives of desired ideal classes, and iterating bounds their elementary divisors by the index of the cyclotomic unit. Passing to the norm tower gives the displayed divisibility, with finite errors removed at height-one primes. The reverse divisibility requires the global index and norm-defect comparison.

For a cyclic group of order $m$ with generator $\sigma$, let $N={\sum }_{i=0}^{m-1}{\sigma }^i$. Subtracting adjacent coefficients and using ${\sigma }^m=1$ proves the displayed group-ring identity. If $p^r\mid m$, then $(\sigma -1)D\equiv -N(\mathrm{mod}{p}^r)$. Together with the norm relations of a root-of-unity Euler system, this makes derivative classes invariant modulo p-power powers and permits Kummer descent, with roots-of-unity corrections where necessary. The congruence axiom controls the associated residue and valuation maps.

A cyclotomic Euler system must relate units at many auxiliary conductors, not just the successive layers of one p-power tower. A useful axiomatization is the root-of-unity Euler system. Fix a finite set of excluded primes $S$ containing $2$, and let $W_S$ be the roots of unity whose orders are prime to every prime in $S$. A system is a nonzero-valued function $\Phi :W_S\to {\overline{\mathbb{Q}}}^{\times }$ with three properties. It is even and Galois equivariant:It satisfies the distribution relation at every prime $\ell \notin S$:Finally, if $\ell \nmid \mathrm{ord}(\xi )$, it satisfies the local congruenceThe values in this congruence are required to be integral at the prime; in the unit example below they are local units. Evenness puts the values in the maximal real cyclotomic subfields. Equivariance identifies conjugates, distribution gives exact norm identities, and the congruence supplies information at newly introduced primes. These are the three distinct inputs used in the descent, rather than a single norm-coherence assumption.

Here is a concrete system extending the units in Q2. Fix a nonzero integer $a$ prime to $p$ and let $S$ contain $2$ and the prime divisors of $a$, but not $p$. PutEvery value is defined and nonzero: the excluded primes make $a$ invertible modulo the order of $\xi$, and that order is odd. The quotient is the same Laurent polynomial $b_a$ used in Q2. Replacing $\xi$ by ${\xi }^{-1}$ changes both numerator and denominator by a minus sign, so it is even; its rational coefficients give Galois equivariance.

For the distribution axiom, $\ell \notin S$ makes $\ell$ odd and prime to $a$. Both exponents $2$ and $2a$ permute ${\mu }_{\ell }$, and ${\prod }_{\rho \in {\mu }_{\ell }}\rho =1$. The calculation in Q2 therefore holds with $p$ replaced by $\ell$:Raising to $p-1$ proves the axiom. At removable points it holds by evaluating the regular Laurent polynomials, so it includes $\xi =1$ and ${\xi }^{\ell }=1$.

For the congruence axiom, a root $\rho$ of order $\ell$ reduces to one at every prime above $\ell$. The integral Laurent polynomial $b_a$ thus gives $b_a(\rho \xi )\equiv b_a(\xi )$. These values are local units when $\xi$ has order prime to $\ell$: its reduction retains that order, and, unless $\xi =1$, both $1-{\xi }^2$ and $1-{\xi }^{2a}$ remain nonzero; at $\xi =1$ the value is the unit $a$. The stated power preserves the congruence. Thus ${\Phi }_a$ satisfies all three Euler-system axioms, and its values at ${\zeta }_n$ are precisely $c_n(a)$. Products, Galois translates, suitable field norms and integral group-ring powers produce the other cyclotomic-unit systems needed in the theory.

The distribution relation explains the Euler factors. If $\ell \nmid m$, $\xi$ has order $m$ and $\rho$ is a primitive $\ell$th root, the conjugates over $\mathbb{Q}({\mu }_m)$ run through the nontrivial $\ell$th roots. Removing the single factor with $\rho =1$ from the distribution product givesWe use arithmetic Frobenius, so ${\mathrm{Fr}}_{\ell }(\xi )={\xi }^{\ell }$. This is the cyclotomic Euler norm factor at a new prime. At a higher power of a prime already dividing the conductor, all $\ell$ translates occur, and the ordinary successive-layer norm relation has no missing-factor quotient. A mere p-power compatible sequence therefore contains only part of the Euler-system information.

To explain the axiomatic method, fix a p-power modulus $P=p^r$ and auxiliary primes $\ell$ with $P\mid \ell -1$, split in the chosen real base field. For the cyclic auxiliary Galois group of order $\ell -1$, choose a generator ${\sigma }_{\ell }$ and defineSubtracting consecutive coefficients gives the concrete identityThis is the Kolyvagin derivative operator for a cyclic group. Apply products of these operators to Euler-system values at squarefree auxiliary conductors. The norm relations control the apparent failure of invariance modulo $P$; Kummer descent, using Hilbert 90 and retaining roots-of-unity corrections where present, turns the derivative data into classes over the base field. They have controlled valuations and are unramified away from the selected auxiliary primes and the excluded set. The local congruence relates the valuation component at a newly added prime to the finite residue-symbol component of the preceding class, with the sign and normalization determined by the chosen generator. This finite-to-singular relation is precisely the additional information not obtainable from norm compatibility alone.

The Chebotarev density theorem supplies auxiliary prime ideals in specified ideal classes while satisfying the splitting and congruence conditions. Reading the controlled valuations as principal-ideal relations successively bounds the p-primary elementary divisors of the ideal class group by the divisibility of the initial cyclotomic-unit class. On suitable nontrivial character components this gives the finite bound $\#A^{\chi }\le \#(E/C{)}^{\chi }$; passing through the norm tower yieldswhere divisibility is between characteristic generators, so the associated ideal containment is reversed. The Euler-system divisibility for real cyclotomic class modules requires the usual local conditions and control of finite errors; it is not a claim that arbitrary compatible units automatically annihilate every class group. The cyclotomic unit index formula, together with the global norm-defect comparison, gives the reverse equality. Q3's exact sequence then proves the main conjecture.

This account separates the theorem inputs from the explicit calculations: the distribution and congruence identities for cyclotomic units are verified above, and the derivative operator is computed directly; the Chebotarev/Kummer descent bounds and the global index comparison are the deeper arithmetic steps. Their usefulness is that the descent uses the axioms rather than a special closed formula for each individual auxiliary unit.

The relevant notion is a root-of-unity Euler system, which contains auxiliary-prime data as well as the p-power tower. Fix a finite set $S$ of primes containing two but not $p$, and let $W_S$ consist of all roots of unity whose orders are prime to every member of $S$. It is a map $\Phi :W_S\to {\overline{\mathbb{Q}}}^{\times }$ satisfying these three axioms:

The first axiom puts $\Phi (\zeta )$ in the maximal real subfield of $\mathbb{Q}(\zeta )$. For $\zeta ={\zeta }_n$, the second axiom is exactly norm compatibility from $K_n^{+}$ to $K_{n-1}^{+}$ for $n\ge 1$. Auxiliary roots with prime orders different from $p$ are essential: mere norm compatibility of a sequence in the tower is not a verification of all three axioms.

The real cyclotomic unit group at a p-power conductor is generated by $-1$ and the symmetric cyclotomic units $c_n(a,b)$ with $p\nmid ab$. Galois conjugates are included, since a conjugate of $c_n(a,b)$ is $c_n(ua,ub)$ for an integer $u$ prime to $p$. Thus a prescribed unit can be expressed aswhere $d_a(z)=z^{-a/2}-z^{a/2}$, with half exponents interpreted in the odd-order root group. Choose $S$ to contain two and every prime divisor of the $a_j$. For $z\ne 1$ in $W_S$ putEach factor at $z\ne 1$ is nonzero because raising $z$ to $a_j$ preserves its order. At $z=1$ each $d_{a_j}$ has a simple zero with first coefficient $-a_j$; the total exponent zero cancels the zeros and signs, giving the stated removable value. Consequently $\Phi ({\zeta }_n)=\epsilon$.

We verify the axioms individually. First, $d_a(z^{-1})=-d_a(z)$; the product's sign is $(-1{)}^{\sum e_j}=1$, so $\Phi$ is even. The unique half-root operation commutes with every field automorphism, all exponents are integers and $\eta$ is rational. Hence $\Phi (\sigma z)=\sigma \Phi (z)$, including at $z=1$.

Second, $q\notin S$ is odd and does not divide any $a_j$. For a variable $x$, choose half powers consistently and useHere $\rho \mapsto {\rho }^a$ permutes ${\mu }_q$, and the product of its half powers is one. Raising these identities to $e_j$ and multiplying gives the distribution identity, since ${\eta }^q=\eta$. If $x=1$ or $x^q=1$, take the removable values after cancellation; the rational-function identity remains valid there. Thus the distribution axiom holds also at the special roots where the uncancelled factors vanish.

Third, fix $q\notin S$, a root $z$ of order prime to $q$, and a prime $\mathfrak{q}$ above $q$. Every $\rho \in {\mu }_q$ reduces to one modulo $\mathfrak{q}$. If $z\ne 1$, its reduction retains its exact order: prime-to-q roots have distinct reductions because their order polynomial is separable in characteristic $q$. Therefore $1-z^{a_j}$ is a local unit, and all the factors and inverse factors are regular at that reduction. Reduction of the expression for $\Phi (\rho z)$ then gives $\Phi (z)$, since the half roots reduce compatibly. This proves the congruence and local integrality for nontrivial $z$.

At $z=1$, write $\rho =1+T$. The cancelled expressionis a unit formal power series over ${\mathbb{Z}}_q$, because every $a_j$ and two are q-adic units. Its constant term is $\eta {\prod }_j{a}_j^{e_j}$. Evaluating at $T=\rho -1$ proves the remaining congruence modulo every prime over $q$. This also covers $\rho =1$. Hence every real cyclotomic unit at any layer is a value of an Cyclotomic Euler system satisfying all three axioms.

For completeness, at a nontrivial root $z$ every ratio $d_a(z)/d_b(z)$ is a global unit. After removing a root-of-unity factor, both it and its inverse are geometric sums because $a,b$ are invertible modulo the order of $z$. Thus the values used at nontrivial roots really are units, not just nonzero algebraic numbers. Values at one may be rational S-units, which is compatible with the definition.

The auxiliary norm factor can also be read off explicitly. If $z$ has order $m>2$ and $q\nmid m$, then for $\rho \ne 1$ of order $q$,Here the arithmetic Frobenius sends $z$ to $z^q$. This convention makes the sign of the Euler factor explicit, while the case of a repeated prime $p$ gives the exact tower norm relation already proved in Question 2.