Principalization (algebra)

In the mathematical field of algebraic number theory, the concept of principalization refers to a situation when, given an extension of algebraic number fields, some ideal (or more generally fractional ideal) of the ring of integers of the smaller field isn't principal but its extension to the ring of integers of the larger field is. Its study has origins in the work of Ernst Kummer on ideal numbers from the 1840s, who in particular proved that for every algebraic number field there exists an extension number field such that all ideals of the ring of integers of the base field (which can always be generated by at most two elements) become principal when extended to the larger field. In 1897 David Hilbert conjectured that the maximal abelian unramified extension of the base field, which was later called the Hilbert class field of the given base field, is such an extension. This conjecture, now known as principal ideal theorem, was proved by Philipp Furtwängler in 1930 after it had been translated from number theory to group theory by Emil Artin in 1929, who made use of his general reciprocity law to establish the reformulation. Since this long desired proof was achieved by means of Artin transfers of non-abelian groups with derived length two, several investigators tried to exploit the theory of such groups further to obtain additional information on the principalization in intermediate fields between the base field and its Hilbert class field. The first contributions in this direction are due to Arnold Scholz and Olga Taussky in 1934, who coined the synonym capitulation for principalization. Another independent access to the principalization problem via Galois cohomology of unit groups is also due to Hilbert and goes back to the chapter on cyclic extensions of number fields of prime degree in his number report, which culminates in the famous Theorem 94.

Extension of classes

Let ${\displaystyle K}$ be an algebraic number field, called the base field, and let ${\displaystyle L/K}$ be a field extension of finite degree. Let ${\displaystyle {\mathcal {O}}_{K},{\mathcal {I}}_{K},{\mathcal {P}}_{K}}$ and ${\displaystyle {\mathcal {O}}_{L},{\mathcal {I}}_{L},{\mathcal {P}}_{L}}$ denote the ring of integers, the group of nonzero fractional ideals and its subgroup of principal fractional ideals of the fields ${\displaystyle K,L}$ respectively. Then the extension map of fractional ideals

$${\displaystyle {\begin{cases}\iota _{L/K}:{\mathcal {I}}_{K}\to {\mathcal {I}}_{L}\\{\mathfrak {a}}\mapsto {\mathfrak {a}}{\mathcal {O}}_{L}\end{cases}}}$$

is an injective group homomorphism. Since ${\displaystyle \iota _{L/K}({\mathcal {P}}_{K})\subseteq {\mathcal {P}}_{L}}$, this map induces the extension homomorphism of ideal class groups

$${\displaystyle {\begin{cases}j_{L/K}:{\mathcal {I}}_{K}/{\mathcal {P}}_{K}\to {\mathcal {I}}_{L}/{\mathcal {P}}_{L}\\{\mathfrak {a}}{\mathcal {P}}_{K}\mapsto ({\mathfrak {a}}{\mathcal {O}}_{L}){\mathcal {P}}_{L}\end{cases}}}$$

If there exists a non-principal ideal ${\displaystyle {\mathfrak {a}}\in {\mathcal {I}}_{K}}$ (i.e. ${\displaystyle {\mathfrak {a}}{\mathcal {P}}_{K}\neq {\mathcal {P}}_{K}}$) whose extension ideal in ${\displaystyle L}$ is principal (i.e. ${\displaystyle {\mathfrak {a}}{\mathcal {O}}_{L}=A{\mathcal {O}}_{L}}$ for some ${\displaystyle A\in {\mathcal {O}}_{L}}$ and ${\displaystyle ({\mathfrak {a}}{\mathcal {O}}_{L}){\mathcal {P}}_{L}=(A{\mathcal {O}}_{L}){\mathcal {P}}_{L}={\mathcal {P}}_{L}}$), then we speak about principalization or capitulation in ${\displaystyle L/K}$. In this case, the ideal ${\displaystyle {\mathfrak {a}}}$ and its class ${\displaystyle {\mathfrak {a}}{\mathcal {P}}_{K}}$ are said to principalize or capitulate in ${\displaystyle L}$. This phenomenon is described most conveniently by the principalization kernel or capitulation kernel, that is the kernel ${\displaystyle \ker(j_{L/K})}$ of the class extension homomorphism.

More generally, let ${\displaystyle {\mathfrak {m}}={\mathfrak {m}}_{0}{\mathfrak {m}}_{\infty }}$ be a modulus in ${\displaystyle K}$, where ${\displaystyle {\mathfrak {m}}_{0}}$ is a nonzero ideal in ${\displaystyle {\mathcal {O}}_{K}}$ and ${\displaystyle {\mathfrak {m}}_{\infty }}$ is a formal product of pair-wise different real infinite primes of ${\displaystyle K}$. Then

$${\displaystyle {\mathcal {S}}_{K,{\mathfrak {m}}}=\langle \alpha {\mathcal {O}}_{K}|\alpha \equiv 1{\bmod {\mathfrak {m}}}\rangle \leq {\mathcal {I}}_{K}({\mathfrak {m}}),}$$

is the ray modulo ${\displaystyle {\mathfrak {m}}}$, where ${\displaystyle {\mathcal {I}}_{K}({\mathfrak {m}})={\mathcal {I}}_{K}({\mathfrak {m}}_{0})}$ is the group of nonzero fractional ideals in ${\displaystyle K}$ relatively prime to ${\displaystyle {\mathfrak {m}}_{0}}$ and the condition ${\displaystyle \alpha \equiv 1{\bmod {\mathfrak {m}}}}$ means ${\displaystyle \alpha \equiv 1{\bmod {{\mathfrak {m}}_{0}}}}$ and ${\displaystyle v(\alpha )>0}$ for every real infinite prime ${\displaystyle v}$ dividing ${\displaystyle {\mathfrak {m}}_{\infty }.}$ Let ${\displaystyle {\mathcal {S}}_{K,{\mathfrak {m}}}\leq {\mathcal {H}}\leq {\mathcal {I}}_{K}({\mathfrak {m}}),}$ then the group ${\displaystyle {\mathcal {I}}_{K}({\mathfrak {m}})/{\mathcal {H}}}$ is called a generalized ideal class group for ${\displaystyle {\mathfrak {m}}.}$ If ${\displaystyle {\mathcal {I}}_{K}({\mathfrak {m}}_{K})/{\mathcal {H}}_{K}}$ and ${\displaystyle {\mathcal {I}}_{L}({\mathfrak {m}}_{L})/{\mathcal {H}}_{L}}$ are generalized ideal class groups such that ${\displaystyle {\mathfrak {a}}{\mathcal {O}}_{L}\in {\mathcal {I}}_{L}({\mathfrak {m}}_{L})}$ for every ${\displaystyle {\mathfrak {a}}\in {\mathcal {I}}_{K}({\mathfrak {m}}_{K})}$ and ${\displaystyle {\mathfrak {a}}{\mathcal {O}}_{L}\in {\mathcal {H}}_{L}}$ for every ${\displaystyle {\mathfrak {a}}\in {\mathcal {H}}_{K}}$, then ${\displaystyle \iota _{L/K}}$ induces the extension homomorphism of generalized ideal class groups:

$${\displaystyle {\begin{cases}j_{L/K}:{\mathcal {I}}_{K}({\mathfrak {m}}_{K})/{\mathcal {H}}_{K}\to {\mathcal {I}}_{L}({\mathfrak {m}}_{L})/{\mathcal {H}}_{L}\\{\mathfrak {a}}{\mathcal {H}}_{K}\mapsto ({\mathfrak {a}}{\mathcal {O}}_{L}){\mathcal {H}}_{L}\end{cases}}}$$

Galois extensions of number fields

Let ${\displaystyle F/K}$ be a Galois extension of algebraic number fields with Galois group ${\displaystyle G=\mathrm {Gal} (F/K)}$ and let ${\displaystyle \mathbb {P} _{K},\mathbb {P} _{F}}$ denote the set of prime ideals of the fields ${\displaystyle K,F}$ respectively. Suppose that ${\displaystyle {\mathfrak {p}}\in \mathbb {P} _{K}}$ is a prime ideal of ${\displaystyle K}$ which does not divide the relative discriminant ${\displaystyle {\mathfrak {d}}={\mathfrak {d}}(F/K)}$, and is therefore unramified in ${\displaystyle F}$, and let ${\displaystyle {\mathfrak {P}}\in \mathbb {P} _{F}}$ be a prime ideal of ${\displaystyle F}$ lying over ${\displaystyle {\mathfrak {p}}}$.

Frobenius automorphism

There exists a unique automorphism ${\displaystyle \sigma \in G}$ such that ${\displaystyle A^{\mathrm {N} ({\mathfrak {p}})}\equiv \sigma (A){\bmod {\mathfrak {P}}}}$ for all algebraic integers ${\displaystyle A\in {\mathcal {O}}_{F}}$, where ${\displaystyle \mathrm {N} ({\mathfrak {p}})}$ is the norm of ${\displaystyle {\mathfrak {p}}}$. The map ${\textstyle \left[{\frac {F/K}{\mathfrak {P}}}\right]:=\sigma }$ is called the Frobenius automorphism of ${\displaystyle {\mathfrak {P}}}$. It generates the decomposition group ${\displaystyle D_{\mathfrak {P}}=\{\sigma \in G|\sigma ({\mathfrak {P}})={\mathfrak {P}}\}}$ of ${\displaystyle {\mathfrak {P}}}$ and its order is equal to the inertia degree ${\displaystyle f:=f({\mathfrak {P}}|{\mathfrak {p}})=[{\mathcal {O}}_{F}/{\mathfrak {P}}:{\mathcal {O}}_{K}/{\mathfrak {p}}]}$ of ${\displaystyle {\mathfrak {P}}}$ over ${\displaystyle {\mathfrak {p}}}$. (If ${\displaystyle {\mathfrak {p}}}$ is ramified then ${\textstyle \left[{\frac {F/K}{\mathfrak {P}}}\right]}$ is only defined and generates ${\displaystyle D_{\mathfrak {P}}}$ modulo the inertia subgroup

$${\displaystyle I_{\mathfrak {P}}=\{\sigma \in G|\sigma (A)\equiv A{\bmod {\mathfrak {P}}}{\text{ for all }}A\in {\mathcal {O}}_{F}\}=\ker(D_{\mathfrak {P}}\to \mathrm {Gal} ({\mathcal {O}}_{F}/{\mathfrak {P}}|{\mathcal {O}}_{K}/{\mathfrak {p}}))}$$

whose order is the ramification index ${\displaystyle e({\mathfrak {P}}|{\mathfrak {p}})}$ of ${\displaystyle {\mathfrak {P}}}$ over ${\displaystyle {\mathfrak {p}}}$). Any other prime ideal of ${\displaystyle F}$ dividing ${\displaystyle {\mathfrak {p}}}$ is of the form ${\displaystyle \tau ({\mathfrak {P}})}$ with some ${\displaystyle \tau \in G}$. Its Frobenius automorphism is given by

$${\displaystyle \left[{\frac {F/K}{\tau ({\mathfrak {P}})}}\right]=\tau \left[{\frac {F/K}{\mathfrak {P}}}\right]\tau ^{-1},}$$

since

$${\displaystyle \tau (A)^{\mathrm {N} ({\mathfrak {p}})}\equiv (\tau \sigma \tau ^{-1})(\tau (A)){\bmod {\tau ({\mathfrak {P}})}}}$$

for all ${\displaystyle A\in {\mathcal {O}}_{F}}$, and thus its decomposition group ${\displaystyle D_{\tau ({\mathfrak {P}})}=\tau D_{\mathfrak {P}}\tau ^{-1}}$ is conjugate to ${\displaystyle D_{\mathfrak {P}}}$. In this general situation, the Artin symbol is a mapping

$${\displaystyle {\mathfrak {p}}\mapsto \left({\frac {F/K}{\mathfrak {p}}}\right):=\left.\left\{\tau \left[{\frac {F/K}{\mathfrak {P}}}\right]\tau ^{-1}\right|\tau \in G\right\}}$$

which associates an entire conjugacy class of automorphisms to any unramified prime ideal ${\displaystyle {\mathfrak {p}}\nmid {\mathfrak {d}}}$, and we have ${\textstyle \left({\frac {F/K}{\mathfrak {p}}}\right)=1}$ if and only if ${\displaystyle {\mathfrak {p}}}$ splits completely in ${\displaystyle F}$.

Factorization of prime ideals

When ${\displaystyle K\subseteq L\subseteq F}$ is an intermediate field with relative Galois group ${\displaystyle H=\mathrm {Gal} (F/L)\leq G}$, more precise statements about the homomorphisms ${\displaystyle \iota _{L/K}}$ and ${\displaystyle j_{L/K}}$ are possible because we can construct the factorization of ${\displaystyle {\mathfrak {p}}}$ (where ${\displaystyle {\mathfrak {p}}}$ is unramified in ${\displaystyle F}$ as above) in ${\displaystyle {\mathcal {O}}_{L}}$ from its factorization in ${\displaystyle {\mathcal {O}}_{F}}$ as follows.^[1][2] Prime ideals in ${\displaystyle {\mathcal {O}}_{F}}$ lying over ${\displaystyle {\mathfrak {p}}}$ are in ${\displaystyle G}$-equivariant bijection with the ${\displaystyle G}$-set of left cosets ${\displaystyle G/D_{\mathfrak {P}}}$, where ${\displaystyle \tau ({\mathfrak {P}})}$ corresponds to the coset ${\displaystyle \tau D_{\mathfrak {P}}}$. For every prime ideal ${\displaystyle {\mathfrak {q}}}$ in ${\displaystyle {\mathcal {O}}_{L}}$ lying over ${\displaystyle {\mathfrak {p}}}$ the Galois group ${\displaystyle H}$ acts transitively on the set of prime ideals in ${\displaystyle {\mathcal {O}}_{F}}$ lying over ${\displaystyle {\mathfrak {q}}}$, thus such ideals ${\displaystyle {\mathfrak {q}}}$ are in bijection with the orbits of the action of ${\displaystyle H}$ on ${\displaystyle G/D_{\mathfrak {P}}}$ by left multiplication. Such orbits are in turn in bijection with the double cosets ${\displaystyle H\backslash G/D_{\mathfrak {P}}}$. Let ${\displaystyle (\tau _{1},\ldots ,\tau _{g})}$ be a complete system of representatives of these double cosets, thus ${\displaystyle G={\dot {\cup }}_{i=1}^{g}\,H\tau _{i}D_{\mathfrak {P}}}$. Furthermore, let ${\displaystyle H\cdot \tau _{i}D_{\mathfrak {P}}}$ denote the orbit of the coset ${\displaystyle \tau _{i}D_{\mathfrak {P}}}$ in the action of ${\displaystyle H}$ on the set of left cosets ${\displaystyle G/D_{\mathfrak {P}}}$ by left multiplication and let ${\displaystyle H\tau _{i}\cdot D_{\mathfrak {P}}}$ denote the orbit of the coset ${\displaystyle H\tau _{i}}$ in the action of ${\displaystyle D_{\mathfrak {P}}}$ on the set of right cosets ${\displaystyle H\backslash G}$ by right multiplication. Then ${\displaystyle {\mathfrak {p}}}$ factorizes in ${\displaystyle {\mathcal {O}}_{L}}$ as ${\textstyle {\mathfrak {p}}{\mathcal {O}}_{L}=\prod _{i=1}^{g}{\mathfrak {q}}_{i}}$, where ${\displaystyle {\mathfrak {q}}_{i}\in \mathbb {P} _{L}}$ for ${\displaystyle 1\leq i\leq g}$ are the prime ideals lying over ${\displaystyle {\mathfrak {p}}}$ in ${\displaystyle L}$ satisfying ${\textstyle {\mathfrak {q}}_{i}{\mathcal {O}}_{F}=\prod _{\varrho }\varrho ({\mathfrak {P}})}$ with the product running over any system of representatives of ${\displaystyle H\cdot \tau _{i}D_{\mathfrak {P}}}$.

We have

$${\displaystyle \#(H\cdot \tau _{i}D_{\mathfrak {P}})\cdot \#D_{\mathfrak {P}}=\#H\tau _{i}D_{\mathfrak {P}}=\#(H\tau _{i}\cdot D_{\mathfrak {P}})\cdot \#H.}$$

Let ${\displaystyle D_{i}}$ be the decomposition group of ${\displaystyle \tau _{i}({\mathfrak {P}})}$ over ${\displaystyle L}$. Then ${\displaystyle D_{i}=H\cap D_{\tau _{i}({\mathfrak {P}})}}$ is the stabilizer of ${\displaystyle \tau _{i}D_{\mathfrak {P}}}$ in the action of ${\displaystyle H}$ on ${\displaystyle G/D_{\mathfrak {P}}}$, so by the orbit-stabilizer theorem we have ${\displaystyle \#D_{i}=\#H/\#(H\cdot \tau _{i}D_{\mathfrak {P}})}$. On the other hand, it's ${\displaystyle \#D_{i}=f(\tau _{i}({\mathfrak {P}})|{\mathfrak {q}}_{i})}$, which together gives

$${\displaystyle f({\mathfrak {q}}_{i}|{\mathfrak {p}})={\frac {f(\tau _{i}({\mathfrak {P}})|{\mathfrak {p}})}{f(\tau _{i}({\mathfrak {P}})|{\mathfrak {q}}_{i})}}={\frac {f({\mathfrak {P}}|{\mathfrak {p}})}{\#D_{i}}}={\frac {\#D_{\mathfrak {P}}}{\#H/\#(H\cdot \tau _{i}D_{\mathfrak {P}})}}={\frac {\#(H\cdot \tau _{i}D_{\mathfrak {P}})\cdot \#D_{\mathfrak {P}}}{\#H}}={\frac {\#H\tau _{i}D_{\mathfrak {P}}}{\#H}}=\#(H\tau _{i}\cdot D_{\mathfrak {P}}).}$$

In other words, the inertia degree ${\displaystyle f_{i}:=f({\mathfrak {q}}_{i}|{\mathfrak {p}})}$ is equal to the size of the orbit of the coset ${\displaystyle H\tau _{i}}$ in the action of ${\textstyle \left[{\frac {F/K}{\mathfrak {P}}}\right]}$ on the set of right cosets ${\displaystyle H\backslash G}$ by right multiplication. By taking inverses, this is equal to the size of the orbit ${\displaystyle D_{\mathfrak {P}}\cdot \tau _{i}^{-1}H}$ of the coset ${\displaystyle \tau _{i}^{-1}H}$ in the action of ${\textstyle \left[{\frac {F/K}{\mathfrak {P}}}\right]}$ on the set of left cosets ${\displaystyle G/H}$ by left multiplication. Also the prime ideals in ${\displaystyle {\mathcal {O}}_{L}}$ lying over ${\displaystyle {\mathfrak {p}}}$ correspond to the orbits of this action.

Consequently, the ideal embedding is given by ${\textstyle \iota _{L/K}({\mathfrak {p}})={\mathfrak {p}}{\mathcal {O}}_{L}=\prod _{i=1}^{g}{\mathfrak {q}}_{i}}$, and the class extension by

$${\displaystyle j_{L/K}({\mathfrak {p}}{\mathcal {H}}_{K})=({\mathfrak {p}}{\mathcal {O}}_{L}){\mathcal {H}}_{L}=\prod _{i=1}^{g}{\mathfrak {q}}_{i}{\mathcal {H}}_{L}.}$$

Artin's reciprocity law

Now further assume ${\displaystyle F/K}$ is an abelian extension, that is, ${\displaystyle G}$ is an abelian group. Then, all conjugate decomposition groups of prime ideals of ${\displaystyle F}$ lying over ${\displaystyle {\mathfrak {p}}}$ coincide, thus ${\displaystyle D_{\mathfrak {p}}:=D_{\tau ({\mathfrak {P}})}}$ for every ${\displaystyle \tau \in G}$, and the Artin symbol ${\textstyle \left({\frac {F/K}{\mathfrak {p}}}\right)=\left[{\frac {F/K}{\mathfrak {P}}}\right]}$ becomes equal to the Frobenius automorphism of any ${\displaystyle {\mathfrak {P}}\mid {\mathfrak {p}}}$ and ${\textstyle A^{\mathrm {N} ({\mathfrak {p}})}\equiv \left({\frac {F/K}{\mathfrak {p}}}\right)(A){\bmod {\mathfrak {P}}}}$ for all ${\displaystyle A\in {\mathcal {O}}_{F}}$ and every ${\displaystyle {\mathfrak {P}}\mid {\mathfrak {p}}}$.

By class field theory,^[3] the abelian extension ${\displaystyle F/K}$ uniquely corresponds to an intermediate group ${\displaystyle {\mathcal {S}}_{K,{\mathfrak {f}}}\leq {\mathcal {H}}\leq {\mathcal {I}}_{K}({\mathfrak {f}})}$ between the ray modulo ${\displaystyle {\mathfrak {f}}}$ of ${\displaystyle K}$ and ${\displaystyle {\mathcal {I}}_{K}({\mathfrak {f}})}$, where ${\displaystyle {\mathfrak {f}}={\mathfrak {f}}_{0}{\mathfrak {f}}_{\infty }={\mathfrak {f}}(F/K)}$ denotes the relative conductor (${\displaystyle {\mathfrak {f}}_{0}}$ is divisible by the same prime ideals as ${\displaystyle {\mathfrak {d}}}$). The Artin symbol

$${\displaystyle {\begin{cases}\mathbb {P} _{K}({\mathfrak {f}})\to G\\{\mathfrak {p}}\mapsto \left({\frac {F/K}{\mathfrak {p}}}\right)\end{cases}}}$$

which associates the Frobenius automorphism of ${\displaystyle {\mathfrak {p}}}$ to each prime ideal ${\displaystyle {\mathfrak {p}}}$ of ${\displaystyle K}$ which is unramified in ${\displaystyle F}$, can be extended by multiplicativity to a surjective homomorphism

$${\displaystyle {\begin{cases}{\mathcal {I}}_{K}({\mathfrak {f}})\to G\\{\mathfrak {a}}=\prod {\mathfrak {p}}^{v_{\mathfrak {p}}({\mathfrak {a}})}\mapsto \left({\frac {F/K}{\mathfrak {a}}}\right):=\prod \left({\frac {F/K}{\mathfrak {p}}}\right)^{v_{\mathfrak {p}}({\mathfrak {a}})}\end{cases}}}$$

with kernel ${\displaystyle {\mathcal {H}}={\mathcal {S}}_{K,{\mathfrak {f}}}\cdot \mathrm {N} _{F/K}({\mathcal {I}}_{F}({\mathfrak {f}}))}$ (where ${\displaystyle {\mathcal {I}}_{F}({\mathfrak {f}})}$ means ${\displaystyle {\mathcal {I}}_{F}({\mathfrak {f}}_{0}{\mathcal {O}}_{F})}$), called Artin map, which induces isomorphism

$${\displaystyle {\begin{cases}{\mathcal {I}}_{K}({\mathfrak {f}})/{\mathcal {H}}\to G=\mathrm {Gal} (F/K)\\{\mathfrak {a}}{\mathcal {H}}\mapsto \left({\frac {F/K}{\mathfrak {a}}}\right)\end{cases}}}$$

of the generalized ideal class group ${\displaystyle {\mathcal {I}}_{K}({\mathfrak {f}})/{\mathcal {H}}}$ to the Galois group ${\displaystyle G}$. This explicit isomorphism is called the Artin reciprocity law or general reciprocity law.^[4]

Group-theoretic formulation of the problem

This reciprocity law allowed Artin to translate the general principalization problem for number fields ${\displaystyle K\subseteq L\subseteq F}$ based on the following scenario from number theory to group theory. Let ${\displaystyle F/K}$ be a Galois extension of algebraic number fields with automorphism group ${\displaystyle G=\mathrm {Gal} (F/K)}$. Assume that ${\displaystyle K\subseteq L\subseteq F}$ is an intermediate field with relative group ${\displaystyle H=\mathrm {Gal} (F/L)\leq G}$ and let ${\displaystyle K'/K,L'/L}$ be the maximal abelian subextension of ${\displaystyle K,L}$ respectively within ${\displaystyle F}$. Then the corresponding relative groups are the commutator subgroups ${\displaystyle G'=\mathrm {Gal} (F/K')\leq G}$, resp. ${\displaystyle H'=\mathrm {Gal} (F/L')\leq H}$. By class field theory, there exist intermediate groups ${\displaystyle {\mathcal {S}}_{K,{\mathfrak {m}}_{K}}\leq {\mathcal {H}}_{K}\leq {\mathcal {I}}_{K}({\mathfrak {d}})}$ and ${\displaystyle {\mathcal {S}}_{L,{\mathfrak {m}}_{L}}\leq {\mathcal {H}}_{L}\leq {\mathcal {I}}_{L}({\mathfrak {d}})}$ such that the Artin maps establish isomorphisms

$${\displaystyle {\begin{aligned}&\left({\frac {K'/K}{\cdot }}\right):{\mathcal {I}}_{K}({\mathfrak {d}})/{\mathcal {H}}_{K}\to \mathrm {Gal} (K'/K)\simeq G/G'\\&\left({\frac {L'/L}{\cdot }}\right):{\mathcal {I}}_{L}({\mathfrak {d}})/{\mathcal {H}}_{L}\to \mathrm {Gal} (L'/L)\simeq H/H'\end{aligned}}}$$

Here ${\displaystyle {\mathfrak {d}}={\mathfrak {d}}(F/K),{\mathcal {I}}_{L}({\mathfrak {d}})}$ means ${\displaystyle {\mathcal {I}}_{L}({\mathfrak {d}}{\mathcal {O}}_{L})}$ and ${\displaystyle {\mathfrak {m}}_{K},{\mathfrak {m}}_{L}}$ are some moduli divisible by ${\displaystyle {\mathfrak {f}}(K'/K),{\mathfrak {f}}(L'/L)}$ respectively and by all primes dividing ${\displaystyle {\mathfrak {d}},{\mathfrak {d}}{\mathcal {O}}_{L}}$ respectively.

The ideal extension homomorphism ${\displaystyle \iota _{L/K}:\,{\mathcal {I}}_{K}({\mathfrak {d}})\to {\mathcal {I}}_{L}({\mathfrak {d}})}$, the induced Artin transfer ${\displaystyle {\tilde {T}}_{G,H}}$ and these Artin maps are connected by the formula

$${\displaystyle {\tilde {T}}_{G,H}\circ \left({\frac {K'/K}{\cdot }}\right)=\left({\frac {L'/L}{\cdot }}\right)\circ \iota _{L/K}.}$$

Since ${\displaystyle {\mathcal {I}}_{K}({\mathfrak {d}})}$ is generated by the prime ideals of ${\displaystyle K}$ which does not divide ${\displaystyle {\mathfrak {d}}}$, it's enough to verify this equality on these generators. Hence suppose that ${\displaystyle {\mathfrak {p}}\in \mathbb {P} _{K}}$ is a prime ideal of ${\displaystyle K}$ which does not divide ${\displaystyle {\mathfrak {d}}}$ and let ${\displaystyle {\mathfrak {P}}\in \mathbb {P} _{F}}$ be a prime ideal of ${\displaystyle F}$ lying over ${\displaystyle {\mathfrak {p}}}$. On the one hand, the ideal extension homomorphism ${\displaystyle \iota _{L/K}}$ maps the ideal ${\displaystyle {\mathfrak {p}}}$ of the base field ${\displaystyle K}$ to the extension ideal ${\displaystyle \iota _{L/K}({\mathfrak {p}})={\mathfrak {p}}{\mathcal {O}}_{L}=\prod _{i=1}^{g}{\mathfrak {q}}_{i}}$ in the field ${\displaystyle L}$, and the Artin map ${\textstyle \left({\frac {L'/L}{\cdot }}\right)}$ of the field ${\displaystyle L}$ maps this product of prime ideals to the product of conjugates of Frobenius automorphisms

$${\displaystyle \prod _{i=1}^{g}\left({\frac {L'/L}{{\mathfrak {q}}_{i}}}\right)=\prod _{i=1}^{g}\left[{\frac {F/L}{\tau _{i}({\mathfrak {P}})}}\right]\cdot H'=\prod _{i=1}^{g}\tau _{i}\left[{\frac {F/L}{\mathfrak {P}}}\right]\tau _{i}^{-1}\cdot H'=\prod _{i=1}^{g}\tau _{i}\left[{\frac {F/K}{\mathfrak {P}}}\right]^{f_{i}}\tau _{i}^{-1}\cdot H',}$$

where the double coset decomposition and its representatives used here is the same as in the last but one section. On the other hand, the Artin map ${\textstyle \left({\frac {K'/K}{\cdot }}\right)}$ of the base field ${\displaystyle K}$ maps the ideal ${\displaystyle {\mathfrak {p}}}$ to the Frobenius automorphism ${\textstyle \left({\frac {K'/K}{\mathfrak {p}}}\right)=\left[{\frac {F/K}{\mathfrak {P}}}\right]\cdot G'}$. The ${\displaystyle g}$-tuple ${\displaystyle (\tau _{1}^{-1},\ldots ,\tau _{g}^{-1})}$ is a system of representatives of double cosets ${\displaystyle D_{\mathfrak {P}}\backslash G/H}$, which correspond to the orbits of the action of ${\textstyle \left[{\frac {F/K}{\mathfrak {P}}}\right]}$ on the set of left cosets ${\displaystyle G/H}$ by left multiplication, and ${\displaystyle f_{i}=\#(H\tau _{i}\cdot D_{\mathfrak {P}})=\#(D_{\mathfrak {P}}\cdot \tau _{i}^{-1}H)}$ is equal to the size of the orbit of coset ${\displaystyle \tau _{i}^{-1}H}$ in this action. Hence the induced Artin transfer maps ${\textstyle \left[{\frac {F/K}{\mathfrak {P}}}\right]\cdot G'}$ to the product

$${\displaystyle {\tilde {T}}_{G,H}\left(\left[{\frac {F/K}{\mathfrak {P}}}\right]\cdot G'\right)=T_{G,H}\left(\left[{\frac {F/K}{\mathfrak {P}}}\right]\right)=\prod _{i=1}^{g}(\tau _{i}^{-1})^{-1}\left[{\frac {F/K}{\mathfrak {P}}}\right]^{f_{i}}\tau _{i}^{-1}\cdot H'=\prod _{i=1}^{g}\tau _{i}\left[{\frac {F/K}{\mathfrak {P}}}\right]^{f_{i}}\tau _{i}^{-1}\cdot H'.}$$

This product expression was the original form of the Artin transfer homomorphism, corresponding to a decomposition of the permutation representation into disjoint cycles.^[5]

Since the kernels of the Artin maps ${\displaystyle \left({\tfrac {K'/K}{\cdot }}\right)}$ and ${\displaystyle \left({\tfrac {L'/L}{\cdot }}\right)}$ are ${\displaystyle {\mathcal {H}}_{K}}$ and ${\displaystyle {\mathcal {H}}_{L}}$ respectively, the previous formula implies that ${\displaystyle \iota _{L/K}({\mathcal {H}}_{K})\subseteq {\mathcal {H}}_{L}}$. It follows that there is the class extension homomorphism ${\displaystyle j_{L/K}:{\mathcal {I}}_{K}({\mathfrak {d}})/{\mathcal {H}}_{K}\to {\mathcal {I}}_{L}({\mathfrak {d}})/{\mathcal {H}}_{L}}$ and that ${\displaystyle j_{L/K}}$ and the induced Artin transfer ${\displaystyle {\tilde {T}}_{G,H}}$ are connected by the commutative diagram in Figure 1 via the isomorphisms induced by the Artin maps, that is, we have equality of two composita ${\displaystyle {\tilde {T}}_{G,H}\circ \left({\tfrac {K'/K}{\cdot }}\right)=\left({\tfrac {L'/L}{\cdot }}\right)\circ j_{L/K}}$.^[3][6]

Class field tower

The commutative diagram in the previous section, which connects the number theoretic class extension homomorphism ${\displaystyle j_{L/K}}$ with the group theoretic Artin transfer ${\displaystyle T_{G,H}}$, enabled Furtwängler to prove the principal ideal theorem by specializing to the situation that ${\displaystyle L=F^{1}(K)}$ is the (first) Hilbert class field of ${\displaystyle K}$, that is the maximal abelian unramified extension of ${\displaystyle K}$, and ${\displaystyle F=F^{2}(K)}$ is the second Hilbert class field of ${\displaystyle K}$, that is the maximal metabelian unramified extension of ${\displaystyle K}$ (and maximal abelian unramified extension of ${\displaystyle F^{1}(K)}$). Then ${\displaystyle K'=L,L'=F,{\mathfrak {d}}={\mathcal {O}}_{K},{\mathcal {H}}_{K}={\mathcal {P}}_{K},{\mathcal {H}}_{L}={\mathcal {P}}_{L}}$ and ${\displaystyle H=G'}$ is the commutator subgroup of ${\displaystyle G}$. More precisely, Furtwängler showed that generally the Artin transfer ${\displaystyle T_{G,G'}}$ from a finite metabelian group ${\displaystyle G}$ to its derived subgroup ${\displaystyle G'}$ is a trivial homomorphism. In fact this is true even if ${\displaystyle G}$ isn't metabelian because we can reduce to the metabelian case by replacing ${\displaystyle G}$ with ${\displaystyle G/G''}$. It also holds for infinite groups provided ${\displaystyle G}$ is finitely generated and ${\displaystyle [G:G']<\infty }$. It follows that every ideal of ${\displaystyle K}$ extends to a principal ideal of ${\displaystyle F^{1}(K)}$.

However, the commutative diagram comprises the potential for a lot of more sophisticated applications. In the situation that ${\displaystyle p}$ is a prime number, ${\displaystyle F=F_{p}^{2}(K)}$ is the second Hilbert p-class field of ${\displaystyle K}$, that is the maximal metabelian unramified extension of ${\displaystyle K}$ of degree a power of ${\displaystyle p,L}$ varies over the intermediate field between ${\displaystyle K}$ and its first Hilbert p-class field ${\displaystyle F_{p}^{1}(K)}$, and ${\displaystyle H=\mathrm {Gal} (F_{p}^{2}(K)/L)\leq G=\mathrm {Gal} (F_{p}^{2}(K)/K)}$ correspondingly varies over the intermediate groups between ${\displaystyle G}$ and ${\displaystyle G'}$, computation of all principalization kernels ${\displaystyle \ker(j_{L/K})}$ and all p-class groups ${\displaystyle \mathrm {Cl} _{p}(L)}$ translates to information on the kernels ${\displaystyle \ker(T_{G,H})}$ and targets ${\displaystyle H/H'}$ of the Artin transfers ${\displaystyle T_{G,H}}$ and permits the exact specification of the second p-class group ${\displaystyle G=\mathrm {Gal} (F_{p}^{2}(K)/K)}$ of ${\displaystyle K}$ via pattern recognition, and frequently even allows to draw conclusions about the entire p-class field tower of ${\displaystyle K}$, that is the Galois group ${\displaystyle \mathrm {Gal} (F_{p}^{\infty }(K)/K)}$ of the maximal unramified pro-p extension ${\displaystyle F_{p}^{\infty }(K)}$ of ${\displaystyle K}$.

These ideas are explicit in the paper of 1934 by A. Scholz and O. Taussky already.^[7] At these early stages, pattern recognition consisted of specifying the annihilator ideals, or symbolic orders, and the Schreier relations of metabelian p-groups and subsequently using a uniqueness theorem on group extensions by O. Schreier.^[8] Nowadays, we use the p-group generation algorithm of M. F. Newman^[9] and E. A. O'Brien^[10] for constructing descendant trees of p-groups and searching patterns, defined by kernels and targets of Artin transfers, among the vertices of these trees.

Galois cohomology

In the chapter on cyclic extensions of number fields of prime degree of his number report from 1897, D. Hilbert^[2] proves a series of crucial theorems which culminate in Theorem 94, the original germ of class field theory. Today, these theorems can be viewed as the beginning of what is now called Galois cohomology. Hilbert considers a finite relative extension ${\displaystyle L/K}$ of algebraic number fields with cyclic Galois group ${\displaystyle G=\mathrm {Gal} (L/K)=\langle \sigma \rangle }$ generated by an automorphism ${\displaystyle \sigma }$ such that ${\displaystyle \sigma ^{\ell }=1}$ for the relative degree ${\displaystyle \ell =[L:K]}$, which is assumed to be an odd prime.

He investigates two endomorphism of the unit group ${\displaystyle U=U_{L}}$ of the extension field, viewed as a Galois module with respect to the group ${\displaystyle G}$, briefly a ${\displaystyle G}$-module. The first endomorphism

$${\displaystyle {\begin{cases}\Delta :U\to U\\E\mapsto E^{\sigma -1}:=\sigma (E)/E\end{cases}}}$$

is the symbolic exponentiation with the difference ${\displaystyle \sigma -1\in \mathbb {Z} [G]}$, and the second endomorphism

$${\displaystyle {\begin{cases}N:U\to U\\E\mapsto E^{T_{G}}:=\prod _{i=0}^{\ell -1}\sigma ^{i}(E)\end{cases}}}$$

is the algebraic norm mapping, that is the symbolic exponentiation with the trace

$${\displaystyle T_{G}=\sum _{i=0}^{\ell -1}\sigma ^{i}\in \mathbb {Z} [G].}$$

In fact, the image of the algebraic norm map is contained in the unit group ${\displaystyle U_{K}}$ of the base field and ${\displaystyle N(E)=\mathrm {N} _{L/K}(E)}$ coincides with the usual arithmetic (field) norm as the product of all conjugates. The composita of the endomorphisms satisfy the relations ${\displaystyle \Delta \circ N=1}$ and ${\displaystyle N\circ \Delta =1}$.

Two important cohomology groups can be defined by means of the kernels and images of these endomorphisms. The zeroth Tate cohomology group of ${\displaystyle G}$ in ${\displaystyle U_{L}}$ is given by the quotient ${\displaystyle H^{0}(G,U_{L}):=\ker(\Delta )/\mathrm {im} (N)=U_{K}/\mathrm {N} _{L/K}(U_{L})}$ consisting of the norm residues of ${\displaystyle U_{K}}$, and the minus first Tate cohomology group of ${\displaystyle G}$ in ${\displaystyle U_{L}}$ is given by the quotient ${\displaystyle H^{-1}(G,U_{L}):=\ker(N)/\mathrm {im} (\Delta )=E_{L/K}/U_{L}^{\sigma -1}}$ of the group ${\displaystyle E_{L/K}=\{E\in U_{L}|N(E)=1\}}$ of relative units of ${\displaystyle L/K}$ modulo the subgroup of symbolic powers of units with formal exponent ${\displaystyle \sigma -1}$.

In his Theorem 92 Hilbert proves the existence of a relative unit ${\displaystyle H\in E_{L/K}}$ which cannot be expressed as ${\displaystyle H=\sigma (E)/E}$, for any unit ${\displaystyle E\in U_{L}}$, which means that the minus first cohomology group ${\displaystyle H^{-1}(G,U_{L})=E_{L/K}/U_{L}^{\sigma -1}}$ is non-trivial of order divisible by ${\displaystyle \ell }$. However, with the aid of a completely similar construction, the minus first cohomology group ${\displaystyle H^{-1}(G,L^{\times })=\{A\in L^{\times }|N(A)=1\}/(L^{\times })^{\sigma -1}}$ of the ${\displaystyle G}$-module ${\displaystyle L^{\times }=L\setminus \{0\}}$, the multiplicative group of the superfield ${\displaystyle L}$, can be defined, and Hilbert shows its triviality ${\displaystyle H^{-1}(G,L^{\times })=1}$ in his famous Theorem 90.

Eventually, Hilbert is in the position to state his celebrated Theorem 94: If ${\displaystyle L/K}$ is a cyclic extension of number fields of odd prime degree ${\displaystyle \ell }$ with trivial relative discriminant ${\displaystyle {\mathfrak {d}}_{L/K}={\mathcal {O}}_{K}}$, which means it's unramified at finite primes, then there exists a non-principal ideal ${\displaystyle {\mathfrak {j}}\in {\mathcal {I}}_{K}\setminus {\mathcal {P}}_{K}}$ of the base field ${\displaystyle K}$ which becomes principal in the extension field ${\displaystyle L}$, that is ${\displaystyle {\mathfrak {j}}{\mathcal {O}}_{L}=A{\mathcal {O}}_{L}\in {\mathcal {P}}_{L}}$ for some ${\displaystyle A\in {\mathcal {O}}_{L}}$. Furthermore, the ${\displaystyle \ell }$th power of this non-principal ideal is principal in the base field ${\displaystyle K}$, in particular ${\displaystyle {\mathfrak {j}}^{\ell }=\mathrm {N} _{L/K}(A){\mathcal {O}}_{K}\in {\mathcal {P}}_{K}}$, hence the class number of the base field must be divisible by ${\displaystyle \ell }$ and the extension field ${\displaystyle L}$ can be called a class field of ${\displaystyle K}$. The proof goes as follows: Theorem 92 says there exists unit ${\displaystyle H\in E_{L/K}\setminus U_{L}^{\sigma -1}}$, then Theorem 90 ensures the existence of a (necessarily non-unit) ${\displaystyle A\in L^{\times }}$ such that ${\displaystyle H=A^{\sigma -1}}$, i. e., ${\displaystyle A^{\sigma }=A\cdot H}$. By multiplying ${\displaystyle A}$ by proper integer if necessary we may assume that ${\displaystyle A}$ is an algebraic integer. The non-unit ${\displaystyle A}$ is generator of an ambiguous principal ideal of ${\displaystyle L/K}$, since ${\displaystyle (A{\mathcal {O}}_{L})^{\sigma }=A^{\sigma }{\mathcal {O}}_{L}=A\cdot H{\mathcal {O}}_{L}=A{\mathcal {O}}_{L}}$. However, the underlying ideal ${\displaystyle {\mathfrak {j}}:=(A{\mathcal {O}}_{L})\cap {\mathcal {O}}_{K}}$ of the subfield ${\displaystyle K}$ cannot be principal. Assume to the contrary that ${\displaystyle {\mathfrak {j}}=\beta {\mathcal {O}}_{K}}$ for some ${\displaystyle \beta \in {\mathcal {O}}_{K}}$. Since ${\displaystyle L/K}$ is unramified, every ambiguous ideal ${\displaystyle {\mathfrak {a}}}$ of ${\displaystyle {\mathcal {O}}_{L}}$ is a lift of some ideal in ${\displaystyle {\mathcal {O}}_{K}}$, in particular ${\displaystyle {\mathfrak {a}}=({\mathfrak {a}}\cap {\mathcal {O}}_{K}){\mathcal {O}}_{L}}$. Hence ${\displaystyle \beta {\mathcal {O}}_{L}={\mathfrak {j}}{\mathcal {O}}_{L}=A{\mathcal {O}}_{L}}$ and thus ${\displaystyle A=\beta E}$ for some unit ${\displaystyle E\in U_{L}}$. This would imply the contradiction ${\displaystyle H=A^{\sigma -1}=(\beta E)^{\sigma -1}=E^{\sigma -1}}$ because ${\displaystyle \beta ^{\sigma -1}=1}$. On the other hand,