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The Kronecker–Weber theorem says that every finite abelian extension of $\mathbb{Q}$ is contained in a cyclotomic field. We prove its quadratic case by identifying the ramification that a quadratic number field must have.

For an odd prime $\ell$, the cyclic Galois group of $\mathbb{Q}({\zeta }_{\ell })/\mathbb{Q}$ has exactly one subgroup of index two. Its fixed field $F_{\ell }$ is quadratic and is unramified at every finite prime other than $\ell$. For a squarefree integer $d$, the discriminant of $\mathbb{Q}(\sqrt{d})$ is $d$ if $d\equiv 1(\mathrm{mod}4)$, and $4d$ otherwise. A nontrivial quadratic discriminant supported on the odd prime $\ell$ must therefore beIndeed $d=\pm 1$ cannot give a nontrivial extension unramified at two, whereas exactly one of $\ell ,-\ell$ is congruent to one modulo four. ThusThis identifies the quadratic subfield of a prime cyclotomic field using ramification, without a Gaussian-sum computation.

Now let $F=\mathbb{Q}(\sqrt{d})$ with $d$ squarefree. Multiply $d$ by ${\prod }_{\ell \mid d,\ell \text{ odd}}{\ell }^{\ast }$. Every odd prime then occurs with even exponent, so its square class is represented by one of $1,-1,2,-2$. Thus adjoining the square roots of the ${\ell }^{\ast }$ removes all possible odd-prime ramification; the remaining quadratic factor is contained inThe original $\sqrt{d}$ is a rational multiple of a product of these square roots. It belongs to the compositum of $\mathbb{Q}({\zeta }_8)$ and the prime cyclotomic fields just used. This compositum is itself contained in a cyclotomic field, proving the theorem for $F$.

The sharper conductor statement follows from the same ramification calculation. For the fundamental discriminant $D$ of $F$, putThe possibilities for $D_2$ follow immediately from the quadratic discriminant formula. The corresponding fields are trivial, $\mathbb{Q}(i)$, $\mathbb{Q}(\sqrt{2})$ and $\mathbb{Q}(\sqrt{-2})$, lying in the cyclotomic fields of conductors $1,4,8,8$. Their conductors are coprime to the odd primes in the product. Hence quadratic Kronecker–Weber theorem by ramification gives the explicit conclusion

Let $F_n=\mathbb{Q}({\mu }_{p^{n+1}})$ and $F_0=\mathbb{Q}({\mu }_p)$. The cyclotomic character identifiesChoose $\gamma$ with cyclotomic value $u=1+p$, and putThe Iwasawa algebra of $G$ is $\Lambda [\Delta ]$. Conjugation, independent of lift as in the preceding solution, makes $X$ a compact Galois module over this ring. It is the p-ramified Iwasawa module, not the unramified class-group module.

Because $p-1$ is a unit in ${\mathbb{Z}}_p$, the Teichmüller character $\omega$ gives orthogonal character idempotentsThey sum to one, so $X={⨁}_{\chi }{X}_{\chi }$, where $X_{\chi }=e_{\chi }X$. Each is a module over the one-variable ring $\Lambda$. Complex conjugation is the element $-1$ of $\Delta$; characters with $\chi (-1)=1$ are even, and those with $\chi (-1)=-1$ are odd.

The basic Iwasawa-module rank theorem in this tower givesIn particular,Thus it would be incorrect to describe all of $X$ as a torsion module.

Here is the class-field unit sequence explaining these ranks. Let $U_{\infty }$ be the inverse limit, under local norms, of the pro-$p$ completions of the unit groups at the unique prime over $p$ in $F_n$. Let ${\overline{E}}_{\infty }$ be the inverse limit of the closures of the global units in these local unit groups, and letunder ideal norms. The Artin reciprocity maps yieldThe final module is the unramified Iwasawa module and is finitely generated torsion, as proved in the other essay. Local-unit theory gives $U_{\infty }$ Iwasawa-module rank $p-1$, with rank one in each $\Delta$-character. The Leopoldt theorem for abelian number fields gives ${\overline{E}}_{\infty }$ rank $(p-1)/2$, with rank one in each even character and zero in each odd character. In odd characters the possible norm-compatible roots of unity contribute a rank-zero term, not a free summand. Subtracting ranks in the exact sequence gives the displayed result and also finite generation.

For any character, the Iwasawa module structure theorem describes $X_{\chi }$ up to pseudo-isomorphism:where the $f_k$ are irreducible distinguished polynomials and the kernel and cokernel of a pseudo-isomorphism are finite. For a torsion component, its characteristic ideal is generated byThe Iwasawa invariants are $\mu =\sum a_j$ and $\lambda =\sum b_k\mathrm{deg}{f}_k$. A characteristic ideal describes the elementary divisors only in aggregate; it is not an assertion that the module is cyclic or that an odd component is actually free.

The trivial character component $X_1$ is zero. Indeed, the maximal abelian pro-$p$ extension of $\mathbb{Q}$ unramified outside $p$ is precisely the rational cyclotomic Zp-extension, by the Kronecker–Weber theorem. Coprime descent using the character idempotent $e_1$, followed by the abelianization over a Zp-extension formula, identifies $(X_1{)}_{\Gamma }$ with the additional abelian pro-$p$ quotient over $\mathbb{Q}$. There is no such extension beyond the cyclotomic one, so $(X_1{)}_{\Gamma }=0$, and the Compact Nakayama lemma gives $X_1=0$.

The central arithmetic description concerns the nontrivial even characters. Let $g_{\chi }(T)$ be the integral Kubota-Leopoldt p-adic L-function power series with conventionFor positive integers $k$ of the matching parity, the interpolation iswhere the Dirichlet character in the Euler factor is its primitive associate, and $B_{k,\eta }$ is a generalized Bernoulli number. The Iwasawa main conjecture, a theorem here, saysThe trivial character is excluded from this displayed analytic normalization: its $p$-adic zeta function has a pole and must be treated separately.

The proof mechanism starts with norm-compatible cyclotomic units, for instance ratiosand their character projections. A Coleman power series encodes a norm-compatible local unit by a single power series. Applying its logarithmic derivative to these cyclotomic units produces the p-adic L-function above. If $C_{\infty }$ denotes their closed norm-limit module, this calculation determinesThe remaining global input isfor nontrivial even $\chi$. In the class-field unit sequence modulo $C_{\infty ,\chi }$, multiplicativity of characteristic ideals cancels these two terms and gives the boxed formula for $X_{\chi }$.

There are two established routes to this global input. The Mazur-Wiles theorem constructs suitable abelian extensions from Galois representations associated with modular forms and the Eisenstein ideal; their sizes supply the missing divisibility. The cyclotomic Euler system route uses norm relations for cyclotomic units at auxiliary primes and descent to bound the class-group module. The analytic class number formula and the cyclotomic-unit index formula then supply the equality of characteristic ideals. These are substantial arithmetic theorems, rather than consequences of the abstract structure theorem alone.

A useful further result is the Ferrero-Washington theorem: $\mu (Y)=0$ for this cyclotomic tower, more generally for cyclotomic towers of abelian number fields. Its proof rules out an identically zero reduction modulo $p$ of the relevant $p$-adic $L$-series, using distribution of $p$-adic digits. Together with the main conjecture, it gives $\mu (X_{\chi })=0$ for each nontrivial even $\chi$. Therefore these torsion components are finitely generated over ${\mathbb{Z}}_p$ up to finite error.

Finally, Kummer reflection in Iwasawa theory relates the even $p$-ramified module to an odd unramified component:where $(1)$ is the Tate twist by the cyclotomic character. The corresponding power-series substitution isThis explains why the $p$-ramified interpolation involves $u^{1-s}$ while the reflected class-group interpolation involves $u^s$. Both the involution and the twist matter. The rank theorem, the class-field sequence, the structure theorem and the explicit even-character characteristic ideals together give the known compact Galois module structure; none justifies replacing every odd component by a free module without further argument.

By the Kronecker–Weber theorem, choose $m$ withRestriction gives a quotientEvery character $\psi \in \hat{G}$ inflates along this quotient and has an associated primitive Dirichlet character ${\chi }_{\psi }$, whose conductor may divide $m$. Comparing Euler factors, or applying the factorization of the Artin $L$-function of the regular representation, gives

The Kronecker–Weber theorem says that every finite abelian extension of $\mathbb{Q}$ lies in some $\mathbb{Q}({\zeta }_m)$.

Indeed, global class field theory assigns to a finite abelian $L/\mathbb{Q}$ the open norm group of an abelian extension $N_{L/\mathbb{Q}}{C}_L$. It contains a congruence subgroup $C_{\mathbb{Q}}(m\infty )$. By part (e), this is the norm group of $\mathbb{Q}({\zeta }_m)$. The inclusion-reversing class-field correspondence therefore gives

For an odd prime $\ell$, the unique quadratic subfield of a prime cyclotomic field has no finite ramification outside $\ell$. The classification of quadratic discriminants identifies it with $\mathbb{Q}(\sqrt{{\ell }^{\ast }})$, where ${\ell }^{\ast }=(-1{)}^{(\ell -1)/2}\ell$. A fundamental discriminant factors as $D=D_2{\prod }_{\ell \mid D,\ell \text{ odd}}{\ell }^{\ast }$ with $D_2\in \{1,-4,8,-8\}$. The remaining quadratic number field is generated by $i$, $\sqrt{2}$, or $\sqrt{-2}$ and lies in $\mathbb{Q}({\zeta }_8)$. Taking the corresponding compositum proves the indicated quadratic case of the Kronecker–Weber theorem.