Cyclotomic unit ($c_n(a)$)

Cyclotomic units include roots of unity and suitable ratios $(1-{\zeta }^a)/(1-\zeta )$ in cyclotomic fields. Compatible roots $\zeta$ make these ratios norm-compatible along cyclotomic towers. Their closed norm-limit module is a concrete submodule of global units, and its local images generate the power series in the Iwasawa main conjecture.

For $1\le k<p$, choose $l$ with $kl\equiv 1(\mathrm{mod}p)$. The geometric sum $1+{\zeta }_p^k+\cdots +{\zeta }_p^{k(l-1)}$ is the inverse of $u_k$. Hence these are units, and their residues cover ${\mathbb{F}}_p^{\times }$. In the cyclotomic Milnor square, this makes the $K_1$ map onto the common field surjective and eliminates its connecting obstruction to $K_0$.

For a real cyclotomic field of odd prime-power conductor, the index of its cyclotomic-unit group has the same p-part as its ideal class number. The analytic class number formula compares the regulator of all units with the explicit cyclotomic-unit regulator. The equality here is stated at the p-part level, so harmless prime-to-p normalization factors do not affect it.

Half exponents mean the inverse of two in the odd-order root-of-unity group. For $p\nmid ab$, the ratio is a real global unit and is compatible under the norms in the p-power tower. Its Coleman power series is $(1+T{)}^{(b-a)/2}(1-(1+T{)}^a)/(1-(1+T{)}^b)$, with constant term $a/b$. Its higher logarithmic derivatives vanish at odd indices and equal $(a^k-b^k)B_k/k$ at even indices, using $B_1=-1/2$.

A cyclotomic Euler system consists of units in auxiliary cyclotomic extensions satisfying compatible norm relations. Reducing at carefully chosen auxiliary primes and descending these relations bounds class-group modules. Combined with global unit-index formulas, this supplies an alternative route to the Iwasawa main conjecture.

Fix a finite excluded set $S$ containing two, and let $W_S$ contain the roots of unity of orders coprime to every prime in $S$. A root-of-unity Euler system is a nonzero algebraic-valued function $\Phi$ on $W_S$ that is even and Galois equivariant, satisfies ${\prod }_{{\rho }^q=1}\Phi (\rho \zeta )=\Phi ({\zeta }^q)$ for $q\notin S$, and satisfies $\Phi (\rho \zeta )\equiv \Phi (\zeta )(\mathrm{mod}\mathfrak{q})$ when the order of $\zeta$ is prime to $q$, for every prime $\mathfrak{q}$ over $q$. Values are required to be locally integral units where the congruence is used. These distribution and congruence properties provide auxiliary cyclotomic extensions, beyond a single norm-compatible p-power sequence.

Let $q\nmid m$ be outside the excluded primes, let $\zeta$ have order $m$, and let $\rho$ be a primitive qth root of unity. Galois equivariance identifies the conjugates with $\Phi ({\rho }^i\zeta )$ for $1\le i\le q-1$. The distribution axiom includes the missing $i=0$ factor, so the field norm is $\Phi ({\zeta }^q)/\Phi (\zeta )$. Arithmetic Frobenius sends $\zeta$ to ${\zeta }^q$, giving the displayed factor. At higher powers of an already ramified prime all q translates occur instead. These are different norm relations; norm compatibility in just a p-power tower does not supply the auxiliary-prime relation by itself.

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.

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.