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A prime ideal of a number field splits completely in a finite extension of degree $n$ when it factors into $n$ distinct prime ideals, each of residue-field degree one. In a Galois extension, an unramified prime ideal splits completely exactly when its Frobenius conjugacy class is the identity class. The Chebotarev density theorem then gives these primes Dirichlet density $1/n$.

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.

Elliptic-unit distribution gives norm-compatible units indexed by auxiliary squarefree ray moduli, with norm factor $1-{\mathrm{Frob}}_{\mathfrak{q}}^{-1}$ at a newly added prime. Kolyvagin derivative operators turn these units into classes with controlled valuations, and suitable auxiliary primes chosen using Chebotarev density theorem bound class-group characteristic ideals. The construction supplies the difficult divisibility in the CM Iwasawa main conjecture.

Let a monic integer polynomial have distinct roots in its splitting field. Outside the finitely many primes dividing its polynomial discriminant or ramifying in that field, its roots reduce to distinct residue roots. The Frobenius automorphism permutes them by $a\mapsto a^p$; a reduced root belongs to ${\mathbb{F}}_p$ exactly when it is fixed. Distinctness of reduction identifies this with a fixed root of the Galois permutation itself. Combining this observation with a derangement in a transitive group action and the Chebotarev density theorem proves that an irreducible polynomial of degree at least two has rootless reduction at a positive Dirichlet density of primes. The degree bound is essential: every monic linear integer polynomial has a root modulo every prime.

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.

Write $F_n=K_n^{+}$, $F_{\infty }={\bigcup }_n{F}_n$, $G=\mathrm{Gal}(F_{\infty }/\mathbb{Q})={\Delta }^{+}\times \Gamma$, where $|{\Delta }^{+}|=(p-1)/2$ and $\Gamma \simeq {\mathbb{Z}}_p$. Set $A={\mathbb{Z}}_p[[G]]$. Choosing a topological generator $\gamma$ of $\Gamma$ identifies its Iwasawa algebra with $\Lambda ={\mathbb{Z}}_p[[S]]$, $S=\gamma -1$; this group variable is distinct from the local-unit interpolation variable $T$. The Teichmüller character decomposes $A$ into a product of copies of $\Lambda$, one for each character of ${\Delta }^{+}$, equivalently each even Teichmüller character of $\mathrm{Gal}(K_0/\mathbb{Q})$. All characteristic-ideal statements below are componentwise in this product.

Let $U_n^1$ be the local principal units at the unique prime above $p$. Let $E_n^1$ and $C_n^1$ be respectively the p-adic closures of global units and cyclotomic units inside $U_n^1$, and take their inverse limits under the field norms. Define $X_{\infty }$ as the Galois group of the maximal abelian pro-p extension of $F_{\infty }$ unramified away from $p$, and $Y_{\infty }$ as that of the maximal everywhere unramified extension. This is the real tower: its prime-ramified module is torsion, by the Leopoldt theorem for abelian number fields. It must not be confused with the positive-rank prime-ramified module of the full complex cyclotomic tower. The unramified Iwasawa torsion theorem gives torsion for $Y_{\infty }$.

A convenient formulation of the Iwasawa main conjecture uses the p-adic zeta pseudomeasure ${\zeta }_p$ on $G$. With the sign conventionits product with the augmentation ideal $I(G)$ is integral and principal. The pole on the trivial-character component is cancelled by that ideal; simply writing an integral quotient by ${\zeta }_p$ would be incorrect. The main assertion in this convention isWe explain the local calculation, the remaining global obstruction, and how the cyclotomic Euler system removes it.

The local calculation is the Iwasawa theorem on local cyclotomic units:One can see why this theorem has exactly this analytic term. For a norm-fixed Coleman power series $f$, its corrected logarithm isThe ratio lies in $1+pR$, so the expression is integral. The norm identity puts its Amice transform in the kernel of the Coleman trace operator, hence gives a measure on ${\mathbb{Z}}_p^{\times }$. Its kth moment is $(1-p^{k-1}){\delta }_k(f)$. The full local exact sequence has p-power roots of unity at its kernel and cokernel. Passing to the real, or even, part removes both, since complex conjugation acts as minus one on their Tate module and $p$ is odd. The corrected logarithm therefore identifies $U_{\infty }^1$ with $A$.

Question 2 now shows that the image of $c(a,b)$ is $([b]-[a]){\zeta }_p$: its even moments are $(1-p^{k-1})(a^k-b^k)B_k/k$. A choice of integer $e$ generating ${\mathbb{Z}}_p^{\times }$ topologically gives a norm-compatible generator $c(e,1)$, after multiplication by its constant Teichmuller lift to make it principal. That constant has zero logarithmic derivatives. Its conjugates generate the closed cyclotomic-unit module, and $[e]-[1]$ generates $I(G)$. This yields the displayed local quotient and its characteristic ideal. The generation statement is an important ingredient; the mere calculation of a few moments would not prove the theorem.

The class-field unit sequence gives the global comparisonMultiplicativity of characteristic ideals in exact sequences givesThus the local theorem proves the main conjecture precisely when one proves $\mathrm{char}(B)=\mathrm{char}(Y_{\infty })$. These terms measure the global-unit and ideal-class obstructions. Discarding them would silently impose a much stronger arithmetic hypothesis.

The first global step is the Euler-system divisibility for real cyclotomic class modules: a generator $f_Y$ of $\mathrm{char}(Y_{\infty })$ divides a generator $f_B$ of $\mathrm{char}(B)$ on every character component. Here is the mechanism. Choose auxiliary primes $q$ splitting completely in a finite layer and satisfying $q\equiv 1(\mathrm{mod}{p}^m)$. For a generator ${\sigma }_q$ of its auxiliary cyclic group, putApply products of these operators to the unit Cyclotomic Euler system of Question 3. The norm axiom and this group-ring identity make the derived elements invariant modulo $p^m$-th powers, so Kummer descent gives classes in $F_n^{\times }/(F_n^{\times }{)}^{p^m}$. The congruence axiom identifies the valuation at each newly introduced prime with a residue symbol of the preceding class; valuations away from the chosen auxiliary primes vanish modulo $p^m$. The Chebotarev density theorem supplies primes in prescribed ideal classes with the required residue-symbol behavior. Adding these primes successively forces the elementary divisors of the class group to divide the available cyclotomic-unit index. Taking the norm limit yields $f_Y\mid f_B$. Finite control defects do not survive localization at height-one primes; using two coprime annihilators of such a finite defect also shows that no hidden p-power factor is left. This is the substantive global argument furnished by the Cyclotomic Euler system, not a consequence of Iwasawa's local theorem alone.

For the reverse comparison use the cyclotomic unit index formula and norm descent, keeping invariants as well as coinvariants. At the base field put $A_0$ for its p-primary ideal class group, and let $N_{\infty }(E_0^1)$ be the universal norm of units in a Zp-extension. Class-field and unit descent, using total ramification and the principal prime over $p$, giveThese are the required norm-defect control statements. The cyclotomic unit index formula, with the p-adic closure justified by Leopoldt, gives $\#A_0=\#(E_0^1/C_0^1)$. Also $B^{\Gamma }=0$: its coinvariants are finite by the displayed control, so its invariants are finite by the Iwasawa module structure theorem, whereas $B$ is a submodule of the cyclic local quotient $H$, which has no nonzero finite submodule. One must retain $Y_{\infty }^{\Gamma }$; dropping it would incorrectly identify all global units with universal norms.

Define the finite Euler characteristic of a one-variable Iwasawa module by ${\chi }_{\Gamma }(M)=\#M_{\Gamma }/\#M^{\Gamma }$. The preceding identities now give an exact cancellation:For a torsion $\Lambda$-module with finite invariant and coinvariant groups, the structure theorem gives ${\chi }_{\Gamma }(M)=|f_M(0){|}_p^{-1}$. Write $f_{B,\chi }=f_{Y,\chi }{h}_{\chi }$ using the Euler-system divisibility. After forgetting ${\Delta }^{+}$, the characteristic series are the products of their character-component series. Equality of these Euler characteristic of a one-variable Iwasawa module values givesEvery summand is nonnegative, since $h_{\chi }\in {\mathbb{Z}}_p[[S]]$. Each is therefore zero. A formal power series with p-adic-unit constant term is a unit, so every $h_{\chi }$ is a unit. Hence $\mathrm{char}(B)=\mathrm{char}(Y_{\infty })$, and cancellation in the class-field exact sequence completes the boxed main conjecture.

The familiar reflected odd class-group formulation follows from Kummer reflection in Iwasawa theory, with the inversion of the group variable and the Tate twist both included. For example, for nontrivial even $\chi$, if $f_{\chi }((1+p{)}^s-1)=L_p(\chi ,s)$, then the p-ramified series is $g_{\chi }(S)=f_{\chi }((1+p)(1+S{)}^{-1}-1)$, and $g_{\chi }((1+p{)}^{1-s}-1)=L_p(\chi ,s)$. The proof above includes the exceptional trivial component through augmentation regularization. It assumes neither a vanishing real class group nor Vandiver's conjecture: the Euler-system divisibility and the norm-defect cancellation replace that extra hypothesis.

A one-variable main conjecture for CM elliptic curves compares a torsion arithmetic Iwasawa module with a measure interpolating special Hecke L-function values. The classical bounded-measure formulation uses a prime of good reduction which splits in the CM field, hence gives ordinary reduction. We first describe the one-prime division-tower formulation, including the analytic construction, and then explain its cyclotomic Selmer counterpart and the change at supersingular primes.

For a precise basic setting, take $E/K$ with complex multiplication by ${\mathcal{O}}_K$, $K$ of class number one, and a split prime $p>3$ of good reduction, prime to the Hecke conductor $\mathfrak{f}$. Write $p{\mathcal{O}}_K=\mathfrak{p}\bar{\mathfrak{p}}$, choose the $p$-adic embedding corresponding to $\mathfrak{p}$, and let $\pi ={\psi }_E(\mathfrak{p})$. Its formal group of an elliptic curve has height one, with $[\pi ](t)\equiv t^p(\mathrm{mod}\mathfrak{p})$. The one-prime division towerhas $\Delta$ of order prime to $p$ and $\Gamma \simeq {\mathbb{Z}}_p$. Its CM character $\kappa :G\to {\mathbb{Z}}_p^{\times }$ describes the action on the prime-primary Tate module. Let $F=F_{\infty }^{\Gamma }$ and $\Lambda =I[[\Gamma ]]\simeq I[[T]]$. We extend coefficients to the complete unramified ring $I$ to include character values and a CM period.

Let $X$ be the Galois group over $F_{\infty }$ of the maximal abelian pro-$p$ extension unramified outside primes above $\mathfrak{p}$. Let $A$ be the analogous everywhere-unramified group, equivalently the norm inverse limit of the $p$-parts of the ideal class groups. Write $U$ for the norm limit of principal local units at $\mathfrak{p}$, $\mathcal{E}$ for the closures of global units in $U$, and $\mathcal{C}$ for the closures of elliptic units. Roots of unity and finite initial-layer errors are retained when formulating integral statements; after passage to a finite-character component they do not change a characteristic ideal.

The class-field unit sequence, obtained by taking norm limits in global Artin reciprocity, givesIn this ordinary one-prime tower, the unit and global-unit terms have the same Iwasawa rank, and the displayed quotients and $X,A$ are torsion on their character components. The equalityis the global-units/class-groups form of the CM main theorem. Multiplicativity of characteristic ideals in the sequence then givesThe analytic construction must identify the latter ideal with a genuinely interpolating p-adic L-function; it is not enough to rename an arbitrary characteristic power series as an L-function.

Here is that construction. For an auxiliary ideal $\mathfrak{a}$ prime to $6p\mathfrak{f}$, a suitably normalized rational elliptic function has divisor a multiple ofTranslate by a generator of the conductor torsion and take the product over its ray-class orbit. This produces a rational function $f_{\mathfrak{a}}(t)$ defined over $K$ which is a unit series at the identity. Evaluating it on compatible ${\mathfrak{p}}^n$-division points gives elliptic units. The reason for the norm relations is geometric: multiplication by $\pi$ groups the translated zeros and poles into fibers, so the product over $E[\mathfrak{p}]$ has the divisor of $f_{\mathfrak{a}}([\pi ]t)$. The normalization fixes the constant, giving the local identityThis is also the norm relation of the Coleman power series for a CM formal group interpolating the resulting local units.

An integral logarithmic correction makes the passage from those units to a bounded measure explicit. In the chosen split completion $K_{\mathfrak{p}}={\mathbb{Q}}_p$, putSince $[\pi ]t\equiv t^p(\mathrm{mod}p)$ and the coefficients of $f$ reduce to ${\mathbb{F}}_p$, the ratio is in $1+p{\mathbb{Z}}_p[[t]]$. For $p>2$, its logarithm divided by $p$ is integral. Constants are treated using the usual $p$-adic logarithm on units. This proves the integrality needed for the Frobenius-corrected logarithm; applying an uncorrected logarithm alone would not prove boundedness.

Choose an isomorphism $\eta :{\hat{\mathbb{G}}}_m\to \hat{E}$ over $I$, and normalize the CM period byThen $A_f(T)={\mathcal{L}}_f(\eta (T))\in I[[T]]$ is the Amice transform of a unique $I$-valued p-adic measure on ${\mathbb{Z}}_p$. The norm relation above givesthe first logarithmic sum is $\log f([\pi ]t)$ and the second is the same sum, because $[\pi ]Q=0$ and there are $p$ points. On the measure side the sum selects $p{\mathbb{Z}}_p$ and multiplies by $p$, so this identity says that the measure is supported on ${\mathbb{Z}}_p^{\times }$. Transport it to $G$ by the CM character and project to the desired component.

The moments can now be calculated rather than postulated. With $D=(1+T)d/dT$,The factor ${\pi }^k$ appears because $[\pi ]$ is multiplication by $\pi$ in the additive formal-logarithm coordinate. The remaining derivative is a Coates–Wiles homomorphism. The complex product for the elliptic function makes it a sum of shifted inverse powers of lattice elements. Grouping these lattice elements by ray-class ideals identifies that sum with $(k-1)!L_{\mathfrak{f}}({\bar{\psi }}_E^k,k)/{\Omega }_{\infty }^k$, multiplied by the auxiliary factor $N\mathfrak{a}-{\psi }_E(\mathfrak{a}{)}^k$. Here $L_{\mathfrak{f}}$ omits primes dividing the original conductor, including when the conductor of the power character becomes smaller.

Normalize the rational functions and matching complex and $p$-adic periods consistently, and remove the auxiliary factors by the distribution relations. They are evaluations of $N\mathfrak{a}-[\mathfrak{a}]$ in the Iwasawa algebra. On each finite-character component an auxiliary ideal can be chosen making this element a unit: its residues at $\mathfrak{p}$ and $\bar{\mathfrak{p}}$ can be prescribed independently by the Chinese remainder theorem. The resulting integral measure ${\mu }_E$ satisfies, on its branch and with $k$ in the corresponding residue class,The exact normalization of periods matters for this equality; multiplication by an integral unit does not affect the ideal in the main conjecture. Finite-order twists are obtained by inserting a character in the ray-class sums, with its local Gauss sum. The lattice calculation proves algebraicity of the normalized complex values, while the integral series construction proves their $p$-adic interpolation. These are different steps.

The Coleman measure map identifies $U_{\chi }$ with a rank-one Iwasawa algebra lattice, up to finite errors, and takes ${\mathcal{C}}_{\chi }$ to the ideal generated by the corresponding power series ${\mathcal{L}}_{\chi }$. The conjecture therefore has the concrete formIts proof uses the Euler system of elliptic units. Adjoining auxiliary ray moduli supplies norm relations with factors $1-{\mathrm{Frob}}_{\mathfrak{q}}^{-1}$. Derivative operators in the auxiliary cyclic groups produce Kolyvagin classes; their controlled valuations relate a new auxiliary prime to its ideal class. Chebotarev density theorem permits the required Frobenius choices. Successively choosing these primes bounds the class-group elementary divisors by the elliptic-unit index, giving the difficult characteristic-ideal divisibility. The opposite comparison comes from the class-number/unit-index formula and passage through the tower. Their equality, combined with the exact sequence above and the logarithmic computation, is the proof mechanism. The ordinary split-prime setting of this formulation is specified in Rubin's one-variable main-conjecture paper.

For completeness, the familiar cyclotomic formulation concerns a different tower. If $E/\mathbb{Q}$ has complex multiplication and good ordinary reduction at $p$, put ${\Gamma }_{\mathrm{cyc}}=\mathrm{Gal}({\mathbb{Q}}_{\infty }/\mathbb{Q})$ andFor a finite layer $L$, its Selmer group of an elliptic curve iswhere $S$ contains the bad primes and primes above $p$; at infinite level take the direct limit. Thus it imposes the local Kummer conditions and sits between rational points tensored with ${\mathbb{Q}}_p/{\mathbb{Z}}_p$ and the $p$-primary Tate–Shafarevich group. Its dual ${\mathcal{X}}_E$ is a torsion Iwasawa module, and the cyclotomic main conjecture isWith compatible reciprocity conventions, the ordinary function has the finite-character interpolationwhere $\alpha$ is the $p$-adic unit root of $X^2-a_{p}X+p$ and $\tau (\chi )={\sum }_{a\mathrm{mod}{p}^n}\chi (a)e^{2\pi ia/p^n}$; characters of this cyclotomic ${\mathbb{Z}}_p$-extension are even. The two-variable CM measure specializes to this line after the CM-character twist. Kummer theory, class-field descriptions of the prime-primary local conditions and control of finite-layer errors identify its algebraic counterpart with the displayed Selmer dual. The one-prime division tower is not itself the cyclotomic tower, so this requires the twist and specialization argument, not just changing its name.

Finally, at a good inert prime the curve has supersingular reduction. The ordinary unit root is absent, the unmodified cyclotomic Selmer dual is not torsion, and the ordinary bounded-function argument does not apply. For CM curves over $\mathbb{Q}$ and odd supersingular $p$, signed local trace conditions and signed functions give instead $\mathrm{char}{\mathcal{X}}_E^{\pm }=(L_p^{\pm }(E))$, as proved in Pollack and Rubin's supersingular main-conjecture theorem. Thus the ordinary hypotheses and the analytic local correction are essential parts of the essay's main-conjecture statement.

We use the Dirichlet density form of the Chebotarev density theorem, the version directly proved by the permitted information at $s=1$. If $L/K$ is a finite Galois extension of number fields with group $G$ and $C$ is a conjugacy class, let $S_C$ be the unramified prime ideals of $K$ whose Frobenius conjugacy class is $C$. ThenThe same conclusion holds for a union of conjugacy classes, replacing $|C|$ by the size of that union. The denominator may equivalently be the sum over all primes of $K$. Here and below $s>1$ is real. This specifies the density being proved; an unweighted prime-counting asymptotic is a stronger formulation and is not being inferred merely from a limit at one.

First establish the necessary pole for a general base field. The Dedekind zeta function as a permutation Artin L-function gives it using exactly the allowed assumptions. Take a finite normal closure $E$ of $K/\mathbb{Q}$, and let $W$ be the permutation representation on ${\mathrm{Hom}}_{\mathbb{Q}}(K,E)$. Its trivial constituent has multiplicity one: an invariant function on the transitive set of embeddings is constant. At a rational prime, a basis of $W^I$ consists of the sums over inertia orbits of embeddings. Frobenius automorphism permutes these orbit sums. Within the decomposition-group orbit belonging to a prime $\mathfrak{p}$ of $K$, the resulting cycle has length $f_{\mathfrak{p}}$, because the residue embeddings are permuted cyclically by finite-field Frobenius. A cycle of length $f$ has determinant $\det (1-T\rho (\mathrm{Frob}))=1-T^f$. Consequently the local factor of an Artin L-function givesincluding at ramified primes. Decomposing $W=1\oplus {⨁}_{\psi \ne 1}{m}_{\psi }\psi$ givesAll factors after $\zeta$ are holomorphic and nonzero at one by the allowed hypothesis. Thus ${\zeta }_K$ has a simple pole there andThis argument also explains why the trivial relative Artin L-function is ${\zeta }_K$, rather than always the Riemann zeta function.

Now let $\chi$ be an irreducible character of $G$, of degree $d_{\chi }$. At an unramified prime the Euler product has local factor $\det (1-{\rho }_{\chi }(g_{\mathfrak{p}})(N\mathfrak{p}{)}^{-s}{)}^{-1}$, where $g_{\mathfrak{p}}$ is a Frobenius representative. The eigenvalues of ${\rho }_{\chi }(g)$ are roots of unity, since $g$ has finite order. Expanding the logarithm for $s>1$ therefore giveswhere the finitely many ramified local factors contribute a function bounded as $s\downarrow 1$. They are nonzero and finite near one because their eigenvalues have modulus one while $(N\mathfrak{p}{)}^{-1}<1$.

The terms $m\ge 2$ are uniformly bounded. Indeed $|\chi (g)|\le d_{\chi }$, and for $N\mathfrak{p}\ge 2$ and $s\ge 1$,There are at most $[K:\mathbb{Q}]$ primes of $K$ above each rational prime and each has norm at least that prime. Hence ${\sum }_{\mathfrak{p}}(N\mathfrak{p}{)}^{-2}<\infty$. Isolating $m=1$ proves the character prime sums in the Chebotarev proof identityFor nontrivial $\chi$, the allowed holomorphy and nonvanishing at one give a local holomorphic logarithm and a bounded right side. The logarithm specified by the Euler series differs from that local branch by a constant multiple of $2\pi i$ on the real interval near one, so it too is bounded. For the trivial character, the right side is $\log (1/(s-1))+O(1)$ by the pole of ${\zeta }_K$. Thus

Expand the conjugacy-class indicator by character orthogonality. For $c\in C$,To see the coefficient, take its inner product with $\chi$: $|G{|}^{-1}{\sum }_{g\in C}\overline{\chi (g)}=|C|\overline{\chi (c)}/|G|$. The irreducible characters form an orthonormal basis of class functions, so this is the full expansion. Substituting the prime-sum estimates leaves only the trivial character's logarithmic term:Division by the diverging logarithm proves the theorem. The same trivial-character calculation shows that all primes of $K$ have density one; adding or removing finitely many primes does not change a Dirichlet density.

There is a necessary qualification for the final polynomial application: the printed claim is false for degree one. For example, $X$ is monic and irreducible in $\mathbb{Z}[X]$, yet has the root zero modulo every prime. We prove the intended assertion for degree $n\ge 2$.

Let $L$ be the splitting field of the monic irreducible polynomial over $\mathbb{Q}$, and let $\Omega$ be its $n$ roots. The Galois group $G$ acts transitively on $\Omega$. There is a derangement in a transitive group action: count the pairs $(g,\alpha )$ satisfying $g\alpha =\alpha$. The orbit-stabilizer theorem givesIf every $g$ fixed at least one root, the identity, which fixes $n\ge 2$, would make this sum at least $|G|+n-1$, a contradiction. Therefore some $g$ fixes no root, and neither does any conjugate of $g$.

For completeness, relate this group action to reduction without assuming the desired splitting result. All roots are algebraic integers, since the polynomial is monic. Exclude the finitely many primes ramifying in $L$ or dividing the nonzero polynomial discriminant. At a prime $\mathfrak{q}$ over any remaining $p$, the roots reduce to distinct elements of $k_{\mathfrak{q}}$ and give all roots of the reduced polynomial. Arithmetic Frobenius permutes them by $b\mapsto b^p$. A reduced root belongs to ${\mathbb{F}}_p$ exactly when it is fixed by this map. Because reduction is injective on the original root set, this happens exactly when the Frobenius permutation fixes an original root. This proves the Frobenius permutation and roots modulo a prime criterion.

The Chebotarev density theorem assigns the conjugacy class of our fixed-point-free $g$ the positive Dirichlet density $|C_g|/|G|$. All but finitely many of these primes have rootless reduction, so there are infinitely many. More precisely, applying the same argument to the union of all fixed-point-free classes gives

We prove the Dirichlet density form of the Chebotarev density theorem. Let $L/K$ be a finite Galois extension with Galois group $G$, let $C$ be a conjugacy class in $G$, and let $S_C$ consist of the unramified prime ideals of $K$ whose Frobenius conjugacy class is $C$. The assertion isThe same result holds for any union of conjugacy classes, by addition. The following proof uses exactly the allowed one-dimensional Artin L-function behavior at $1$ and the permitted induction and Euler-product properties.

First determine the order of vanishing at $1$ for an arbitrary character of a representation $\chi$ of $G$. The Brauer induction theorem supplies subgroups $H_j\le G$, linear characters ${\psi }_j$ of $H_j$, and integers $n_j$, possibly negative, such thatPut $K_j=L^{H_j}$. Compatibility of Artin L-functions with induced representations and direct sums gives, initially for $\mathrm{Re}s>1$,Each ${\psi }_j$ is a one-dimensional Galois representation for $L/K_j$, so the stated analytic assumptions apply to every factor. A trivial ${\psi }_j$ contributes a simple pole; a nontrivial ${\psi }_j$ is holomorphic and nonzero at one. Thus the product is meromorphic near one andThe character inner product is $⟨\alpha ,\beta {⟩}_G=|G{|}^{-1}{\sum }_{g\in G}\alpha (g)\overline{\beta (g)}$. Applying Frobenius reciprocity to each induced term givesIn particular, a nontrivial irreducible character has order zero, so its Artin L-function is holomorphic and nonzero near one. For the trivial character, $L_K(1_G,s)={\zeta }_K(s)$, the Dedekind zeta function of the base field $K$, and its pole is simple. This argument establishes the behavior needed at one, not holomorphy everywhere for arbitrary Artin L-functions.

Next convert that order calculation into a prime sum. For an irreducible character $\chi$, choose a unitary representation of dimension $d_{\chi }$ affording it, and choose a Frobenius automorphism $g_{\mathfrak{p}}$ at each unramified prime. The logarithm specified by the absolutely convergent Euler product gives, for real $s>1$,Only finitely many ramified primes occur. Their Euler-factor logarithms $R_{\chi }(s)$ stay bounded near one, since the relevant eigenvalues have modulus one and $(N\mathfrak{p}{)}^{-1}<1$. Also $|\chi (g)|\le d_{\chi }$, so the contribution with $m\ge 2$ is uniformly bounded for $s\ge 1$ byConvergence follows, for example, from the convergent ideal series defining ${\zeta }_K(2)$. ThereforeWriting $a_{\chi }=⟨\chi ,1_G{⟩}_G$, the order computation means $L_K(\chi ,s)=(s-1{)}^{-a_{\chi }}{U}_{\chi }(s)$, where $U_{\chi }$ is holomorphic and nonzero near one. Choose its local logarithm. Along the real interval approaching one, this and the Euler-product logarithm differ only by a fixed multiple of $2\pi i$. ConsequentlyOnly the trivial irreducible character has $a_{\chi }=1$; all the others have $a_{\chi }=0$.

Finally, character orthogonality makes the irreducible characters an orthonormal basis of class functions. For $c\in C$, the coefficient of $\chi$ in the class indicator is $⟨1_C,\chi {⟩}_G=(|C|/|G|)\overline{\chi (c)}$. HenceSum this identity over the unramified primes with weight $(N\mathfrak{p}{)}^{-s}$. The finite character sum and the boxed estimates giveDividing by $\log (1/(s-1))$ proves the asserted Dirichlet density. The same Euler-product calculation with ${\zeta }_K$ gives ${\sum }_{\mathfrak{p}}(N\mathfrak{p}{)}^{-s}=\log (1/(s-1))+O(1)$, so using all primes as the denominator is equivalent. Finitely many ramified primes do not affect either limit. No prime-counting asymptotic or natural-density Tauberian step is required by this proof.

A set $S$ of prime numbers has Dirichlet density $\delta$ if the following limit exists:Equivalently, the denominator can be ${\sum }_p{p}^{-s}$, since that sum is $\log (1/(s-1))+O(1)$. Adding or removing finitely many prime numbers leaves the Dirichlet density unchanged.

Put $\alpha =\sqrt[4]{2}>0$ and $L=\mathbb{Q}(\alpha ,i)$. This is the splitting field of $X^4-2$, whose four roots of a polynomial are $\alpha ,i\alpha ,-\alpha ,-i\alpha$. The polynomial is an Eisenstein polynomial at $2$, so $[\mathbb{Q}(\alpha ):\mathbb{Q}]=4$. Since $\mathbb{Q}(\alpha )$ is contained in the real numbers, it does not contain $i$, giving $[L:\mathbb{Q}]=8$. Thus $L/\mathbb{Q}$ is a Galois extension of degree $8$. Its Galois group is the dihedral group of order $8$: the field automorphisms $r(\alpha )=i\alpha$, $r(i)=i$ and $s(\alpha )=\alpha$, $s(i)=-i$ satisfy $r^4=s^2=1$ and $srs=r^{-1}$.

For an odd prime number $p$, the condition $p\equiv 1(\mathrm{mod}4)$ means that the finite field ${\mathbb{F}}_p$ contains all fourth roots of unity. Under this condition, $2$ is a quartic residue exactly when $X^4-2$ has a root in ${\mathbb{F}}_p$; multiplying that root by $1,i,-1,-i$ gives all four distinct roots. Conversely, complete splitting of $X^4-2$ over ${\mathbb{F}}_p$ gives both a fourth root of $2$ and a primitive fourth root of unity, so it also forces $p\equiv 1(\mathrm{mod}4)$.

The polynomial discriminant of $X^4-2$ is $-2^{11}$. Hence every odd prime number is unramified in $L$, and the root-splitting criterion is equivalent to its Frobenius automorphism acting trivially on all the roots. Since the roots generate $L$, this is equivalent to the Frobenius automorphism being the identity, or to $p$ being a completely split prime of $L/\mathbb{Q}$. The Chebotarev density theorem says that the unramified prime numbers with Frobenius conjugacy class $C$ have Dirichlet density $|C|/|\mathrm{Gal}(L/\mathbb{Q})|$. Here $C=\{1\}$, so

The Chebotarev density theorem says that if $L/K$ is a finite Galois extension with group $G$ and $C\subseteq G$ is a conjugacy class, then the unramified primes $\mathfrak{p}$ whose Frobenius conjugacy class equals $C$ have Dirichlet density

A prime splits completely in $L$ exactly when its Frobenius conjugacy class is $\{1\}$. Applying the Chebotarev density theorem to the identity class gives

Let ${\mathfrak{f}}_{L/K}$ be the conductor of an abelian extension and let $\mathfrak{m}$ be divisible by it. For any $\sigma \in \mathrm{Gal}(L/K)$, the Chebotarev density theorem supplies infinitely many unramified prime ideals $\mathfrak{p}$ withOnly finitely many primes divide ${\mathfrak{m}}_0$, so $\mathfrak{p}$ may be chosen prime to ${\mathfrak{m}}_0$. It then belongs to $I_K(\mathfrak{m})$ and its image under the Artin reciprocity map is $\sigma$. Hence the Artin map is surjective.

The Chebotarev density theorem states that for a finite Galois extension $L/K$ and a conjugacy class $C\subseteq \mathrm{Gal}(L/K)$, the unramified primes whose Frobenius conjugacy class is $C$ have Dirichlet density

A prime ideal of $K$ is principal exactly when its Artin symbol inis the identity. The group has order two. Applying the Chebotarev density theorem to the identity conjugacy class gives density