Principalization (algebra)

In the mathematical field of algebraic number theory, the concept principalization has its origin in David Hilbert's 1897 conjecture that all ideals of an algebraic number field, which can always be generated by two algebraic numbers, become principal ideals, generated by a single algebraic number, when they are transferred to the maximal abelian unramified extension field, which was later called the Hilbert class field, of the given base field. More than thirty years later, Philipp Furtwängler succeeded in proving this principal ideal theorem in 1930, after it had been translated from number theory to group theory by E. 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 A. Scholz and O. 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 relative extensions of prime degree in his number report 1897, which culminates in the famous Theorem 94.

Extension of classes

Let $K$ be an algebraic number field, called the base field, and let $L\vert K$ be a field extension of finite degree.

Definition.

The embedding monomorphism of fractional ideals $\iota_{L\vert K}:\ \mathcal{I}_K\to\mathcal{I}_L,\ \mathfrak{a}\mapsto\mathfrak{a}\mathcal{O}_L$ , where $\mathcal{O}_L$ denotes the ring of integers of $L$ , induces the extension homomorphism of ideal classes $j_{L\vert K}:\ \mathcal{I}_K/\mathcal{P}_K\to\mathcal{I}_L/\mathcal{P}_L,\ \mathfrak{a}\mathcal{P}_K\mapsto(\mathfrak{a}\mathcal{O}_L)\mathcal{P}_L$ , where $\mathcal{P}_K$ and $\mathcal{P}_L$ denote the subgroups of principal ideals.

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

Remark.

When $F$ is a Galois extension of $K$ with automorphism group $G=\mathrm{Gal}(F\vert K)$ such that $K\le L\le F$ is an intermediate field with relative group $H=\mathrm{Gal}(F\vert L)\le G$ , more precise statements about the homomorphisms $\iota_{L\vert K}$ and $j_{L\vert K}$ are possible by using group theory. According to the theories of A. Hurwitz 1895 ^[1] and D. Hilbert 1897 ^[2] on the decomposition of a prime ideal $\mathfrak{p}\in\mathbb{P}_K$ in the extension $L\vert K$ , viewed as a subextension of $F\vert K$ , we have $\mathfrak{p}\mathcal{O}_L=\prod_{i=1}^g\,\mathfrak{q}_i$ , where the $\mathfrak{q}_i=\prod_{\varrho\in H\tau_iD}\,\varrho(\mathfrak{P})\in\mathbb{P}_L$ , with $1\le i\le g$ , are the prime ideals lying over $\mathfrak{p}$ in $L$ , expressed by a fixed prime ideal $\mathfrak{P}\in\mathbb{P}_F$ dividing $\mathfrak{p}$ in $F$ and a double coset decomposition $G=\dot{\cup}_{i=1}^g\,H\tau_iD$ of $G$ modulo $H$ and modulo the decomposition group (stabilizer) $D=\lbrace\sigma\in G\mid\sigma(\mathfrak{P})=\mathfrak{P}\rbrace$ of $\mathfrak{P}$ in $G$ , with a complete system of representatives $(\tau_1,\ldots,\tau_g)$ . The order of the decomposition group $D$ is the inertia degree $f(\mathfrak{P}\vert\mathfrak{p})$ of $\mathfrak{P}$ over $K$ .

Consequently, the ideal embedding is given by $\iota_{L\vert K}(\mathfrak{p})=\mathfrak{p}\mathcal{O}_L=\prod_{i=1}^g\,\mathfrak{q}_i$ , and the class extension by $j_{L\vert K}(\mathfrak{p}\mathcal{P}_K)=(\mathfrak{p}\mathcal{O}_L)\mathcal{P}_L=\prod_{i=1}^g\,\mathfrak{q}_i\mathcal{P}_L$ .

Artin's reciprocity law

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

Then, there exists a unique automorphism $\sigma\in G$ such that $A^{\mathrm{Norm}_{K\vert\mathbb{Q}}(\mathfrak{p})}\equiv\sigma(A)\pmod{\mathfrak{P}}$ , for all algebraic integers $A\in\mathcal{O}_F$ , which is called the Frobenius automorphism $\left\lbrack\frac{F\vert K}{\mathfrak{P}}\right\rbrack:=\sigma$ of $\mathfrak{P}$ and generates the cyclic decomposition group $D_{\mathfrak{P}}=\langle\sigma\rangle$ of $\mathfrak{P}$ . Any other prime ideal of $F$ dividing $\mathfrak{p}$ is of the form $\tau(\mathfrak{P})$ with some $\tau\in G$ . Its Frobenius automorphism is given by $\left\lbrack\frac{F\vert K}{\tau(\mathfrak{P})}\right\rbrack=\tau\left\lbrack\frac{F\vert K}{\mathfrak{P}}\right\rbrack\tau^{-1}$ , since $\tau(A)^{\mathrm{Norm}_{K\vert\mathbb{Q}}(\mathfrak{p})}\equiv(\tau\sigma\tau^{-1})(\tau(A))\pmod{\tau(\mathfrak{P})}$ , for all $A\in\mathcal{O}_F$ , and thus its decomposition group $D_{\tau(\mathfrak{P})}=\tau D_{\mathfrak{P}}\tau^{-1}$ is conjugate to $D_{\mathfrak{P}}$ . In this general situation, the Artin symbol is a mapping $\mathfrak{p}\mapsto\left(\frac{F\vert K}{\mathfrak{p}}\right):=\lbrace\tau\left\lbrack\frac{F\vert K}{\mathfrak{P}}\right\rbrack\tau^{-1}\mid\tau\in G\rbrace$ which associates an entire conjugacy class of automorphisms to any unramified prime ideal $\mathfrak{p}\not\vert\mathfrak{d}$ , and we have $\left(\frac{F\vert K}{\mathfrak{p}}\right)=1$ if and only if $\mathfrak{p}$ splits completely in $F$ .

Now let $F\vert K$ be an abelian extension, that is, the Galois group $G=\mathrm{Gal}(F\vert K)$ is an abelian group. Then, all conjugate decomposition groups of prime ideals of $F$ lying over $\mathfrak{p}$ coincide $D_{\tau(\mathfrak{P})}=:D_{\mathfrak{p}}$ , for any $\tau\in G$ , and the Artin symbol $\left(\frac{F\vert K}{\mathfrak{p}}\right)=\left\lbrack\frac{F\vert K}{\mathfrak{P}}\right\rbrack$ becomes equal to the Frobenius automorphism of any $\mathfrak{P}\mid\mathfrak{p}$ , since $A^{\mathrm{Norm}_{K\vert\mathbb{Q}}(\mathfrak{p})}\equiv\left(\frac{F\vert K}{\mathfrak{p}}\right)(A)\pmod{\mathfrak{p}}$ , for all $A\in\mathcal{O}_F$ .

By class field theory, ^[3] the abelian extension $F\vert K$ uniquely corresponds to an intermediate group $\mathcal{S}_{K,\mathfrak{f}}\le\mathcal{H}\le\mathcal{P}_K(\mathfrak{f})$ between the ray modulo $\mathfrak{f}$ of $K$ , that is $\mathcal{S}_{K,\mathfrak{f}}=\lbrace\alpha\mathcal{O}_K\mid\alpha\equiv 1\pmod{\mathfrak{f}}\rbrace$ , and the group of principal ideals coprime to $\mathfrak{f}$ of $K$ , where $\mathfrak{f}=\mathfrak{f}(F\vert K)$ denotes the relative conductor. (Note that $\mathfrak{p}\mid\mathfrak{f}$ if and only if $\mathfrak{p}\mid\mathfrak{d}$ , but $\mathfrak{f}$ is minimal with this property.) The Artin symbol $\mathbb{P}_K(\mathfrak{f})\to G,\ \mathfrak{p}\mapsto\left(\frac{F\vert K}{\mathfrak{p}}\right)$ , which associates the Frobenius automorphism of $\mathfrak{p}$ to each prime ideal $\mathfrak{p}$ of $K$ which is unramified in $F$ , can be extended by multiplicativity to an epimorphism $\mathcal{I}_K(\mathfrak{f})\to G,\ \mathfrak{a}=\prod\,\mathfrak{p}^{v_{\mathfrak{p}}(\mathfrak{a})}\mapsto\left(\frac{F\vert K}{\mathfrak{a}}\right):=\prod\,\left(\frac{F\vert K}{\mathfrak{p}}\right)^{v_{\mathfrak{p}}(\mathfrak{a})}$ with kernel $\mathcal{H}=\mathcal{S}_{K,\mathfrak{f}}\cdot\mathrm{Norm}_{F\vert K}(\mathcal{I}_F(\mathfrak{f}))$ , which induces the Artin isomorphism, or Artin map, $\mathcal{I}_K(\mathfrak{f})/\mathcal{H}\to G=\mathrm{Gal}(F\vert K),\ \mathfrak{a}\mathcal{H}\mapsto\left(\frac{F\vert K}{\mathfrak{a}}\right)$ of the generalized ideal class group $\mathcal{I}_K(\mathfrak{f})/\mathcal{H}$ to the Galois group $G$ , which maps the class $\mathfrak{a}\mathcal{H}$ of $\mathfrak{a}$ to the Artin symbol $\left(\frac{F\vert K}{\mathfrak{a}}\right)$ of $\mathfrak{a}$ . This explicit isomorphism is called the Artin reciprocity law or general reciprocity law. ^[4]

Figure 1: Commutative diagram connecting the class extension with the Artin transfer.

Commutative diagram

E. Artin's translation of the general principalization problem for a number field extension $L\vert K$ from number theory to group theory is based on the following scenario. Let $F\vert K$ be a Galois extension of algebraic number fields with automorphism group $G=\mathrm{Gal}(F\vert K)$ . Suppose that $\mathfrak{p}\in\mathbb{P}_K$ is a prime ideal of $K$ which does not divide the relative discriminant $\mathfrak{d}=\mathfrak{d}(F\vert K)$ , and is therefore unramified in $F$ , and let $\mathfrak{P}\in\mathbb{P}_F$ be a prime ideal of $F$ lying over $\mathfrak{p}$ . Assume that $K\le L\le F$ is an intermediate field with relative group $H=\mathrm{Gal}(F\vert L)\le G$ and let $K^\prime\vert K$ , resp. $L^\prime\vert L$ , be the maximal abelian subextension of $K$ , resp. $L$ , within $F$ . Then, the corresponding relative groups are the commutator subgroups $G^\prime=\mathrm{Gal}(F\vert K^\prime)\le G$ , resp. $H^\prime=\mathrm{Gal}(F\vert L^\prime)\le H$ .

By class field theory, there exist intermediate groups $\mathcal{S}_{K,\mathfrak{d}}\le\mathcal{H}_K\le\mathcal{P}_K(\mathfrak{d})$ and $\mathcal{S}_{L,\mathfrak{d}}\le\mathcal{H}_L\le\mathcal{P}_L(\mathfrak{d})$ such that the Artin maps establish isomorphisms $\left(\frac{K^\prime\vert K}{\cdots}\right):\,\mathcal{I}_K(\mathfrak{d})/\mathcal{H}_K\to\mathrm{Gal}(K^\prime\vert K)\simeq G/G^\prime$ and $\left(\frac{L^\prime\vert L}{\cdots}\right):\,\mathcal{I}_L(\mathfrak{d})/\mathcal{H}_L\to\mathrm{Gal}(L^\prime\vert L)\simeq H/H^\prime$ .

The class extension homomorphism $j_{L\vert K}$ and the Artin transfer, more precisely, the induced transfer $\tilde{T}_{G,H}$ , are connected by the commutative diagram in Figure 1 via these Artin isomorphisms, that is, we have equality of two composita $\tilde{T}_{G,H}\circ\left(\frac{K^\prime\vert K}{\cdots}\right)=\left(\frac{L^\prime\vert L}{\cdots}\right)\circ j_{L\vert K}$ . ^[5] The justification for this statement consists in analyzing the two paths of composite mappings. ^[3] On the one hand, the class extension homomorphism $j_{L\vert K}$ maps the generalized ideal class $\mathfrak{p}\mathcal{H}_K$ of the base field $K$ to the extension class $j_{L\vert K}(\mathfrak{p}\mathcal{H}_K)=(\mathfrak{p}\mathcal{O}_L)\mathcal{H}_L=\prod_{i=1}^g\,\mathfrak{q}_i\mathcal{H}_L$ in the field $L$ , and the Artin isomorphism $\left(\frac{L^\prime\vert L}{\cdots}\right)$ of the field $L$ maps this product of classes of prime ideals to the product of conjugates of Frobenius automorphisms $\prod_{i=1}^g\,\left(\frac{L^\prime\vert L}{\mathfrak{q}_i}\right)=\prod_{i=1}^g\,\left\lbrack\frac{F\vert L}{\tau_i(\mathfrak{P})}\right\rbrack\cdot H^\prime=\prod_{i=1}^g\,\tau_i\left\lbrack\frac{F\vert K}{\mathfrak{P}}\right\rbrack^{f_i}\tau_i^{-1}\cdot H^\prime$ . Here, the double coset decomposition and its representatives were used, in perfect analogy to the last but one section. On the other hand, the Artin isomorphism $\left(\frac{K^\prime\vert K}{\cdots}\right)$ of the base field $K$ maps the generalized ideal class $\mathfrak{p}\mathcal{H}_K$ to the Frobenius automorphism $\left(\frac{K^\prime\vert K}{\mathfrak{p}}\right)$ , and the induced Artin transfer maps the symbol $\left(\frac{K^\prime\vert K}{\mathfrak{p}}\right)=\left\lbrack\frac{F\vert K}{\mathfrak{P}}\right\rbrack\cdot G^\prime$ to the product $\tilde{T}_{G,H}\left(\left\lbrack\frac{F\vert K}{\mathfrak{P}}\right\rbrack\cdot G^\prime\right)=\prod_{i=1}^g\,\tau_i\left\lbrack\frac{F\vert K}{\mathfrak{P}}\right\rbrack^{f_i}\tau_i^{-1}\cdot H^\prime$ . ^[6] This product expression was the original form of the Artin transfer homomorphism, corresponding to a decomposition of the permutation representation into disjoint cycles.

Class field tower

The commutative diagram in the previous section, which connects the number theoretic class extension homomorphism $j_{L\vert K}$ with the group theoretic Artin transfer $T_{G,H}$ , enabled Furtwängler to prove the principal ideal theorem by specializing to the situation that $L=F^1(K)$ is the first Hilbert class field of $K$ , that is the maximal abelian unramified extension of $K$ , and $H=G^\prime$ is the commutator subgroup of $G$ . More precisely, Furtwängler showed that generally the Artin transfer $T_{G,G^\prime}$ from a metabelian group $G$ to its derived subgroup $G^\prime$ maps all elements of $G$ to the neutral element of $G^\prime$ .

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

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 relative extensions of prime degree of his number report 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 $L\vert K$ of algebraic number fields with cyclic Galois group $G=\mathrm{Gal}(L\vert K)=\langle\sigma\rangle$ generated by an automorphism $\sigma$ such that $\sigma^\ell=1$ for the relative degree $\ell=\lbrack L:K\rbrack$ , which is assumed to be an odd prime.

He investigates two endomorphism of the unit group $U=U_L$ of the extension field, viewed as a Galois module with respect to the group $G$ , briefly a $G$ -module. The first endomorphism $\Delta:\,U\to U,\ E\mapsto E^{\sigma-1}:=\sigma(E)/E$ is the symbolic exponentiation with the difference $\sigma-1\in\mathbb{Z}\lbrack G\rbrack$ , and the second endomorphism $N:\,\,U\to U,\ E\mapsto E^{T_G}:=\prod_{i=0}^{\ell-1}\,\sigma^i(E)$ is the algebraic norm mapping, that is the symbolic exponentiation with the trace $T_G=\sum_{i=0}^{\ell-1}\,\sigma^i\in\mathbb{Z}\lbrack G\rbrack$ . In fact, the image of the algebraic norm map is contained in the unit group $U_K$ of the base field, and thus $N(E)=\mathrm{N}_{L\vert K}(E)$ coincides with the usual arithmetic norm as the product of all conjugates. The composita of the endomorphisms satisfy the relations $\Delta\circ N=1$ and $N\circ\Delta=1$ .

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

In his Theorem 92, under the additional assumption that $L\vert K$ be unramified, Hilbert proves the existence of a relative unit $H\in E_{L\vert K}$ which cannot be expressed as $H=\sigma(E)/E$ , for any unit $E\in U_L$ , which means that the minus first cohomology group $H^{-1}(G,U_L)=E_{L\vert K}/U_L^{\sigma-1}$ is non-trivial of order divisible by $\ell$ . However, with the aid of a completely similar construction, the minus first cohomology group $H^{-1}(G,L^{\times})=\lbrace A\in L^{\times}\mid N(A)=1\rbrace/(L^{\times})^{\sigma-1}$ of the $G$ -module $L^{\times}=L\setminus\lbrace 0\rbrace$ , the multiplicative group of the superfield $L$ , can be defined, and Hilbert shows its triviality $H^{-1}(G,L^{\times})=1$ in Theorem 90. Applied to the particular unit $H\in E_{L\vert K}\setminus U_L^{\sigma-1}$ , this ensures the existence of a non-unit $A\in L^{\times}$ such that $H=A^{\sigma-1}$ , i. e., $A^{\sigma}=A\cdot H$ .

The non-unit $A$ is generator of an ambiguous principal ideal of $L\vert K$ , since $(A\mathcal{O}_L)^{\sigma}=A^{\sigma}\mathcal{O}_L=A\cdot H\mathcal{O}_L=A\mathcal{O}_L$ . However, the underlying ideal $\mathfrak{j}:=(A\mathcal{O}_L)\cap\mathcal{O}_K$ of the subfield $K$ cannot be principal, because otherwise we had $\mathfrak{j}=\beta\mathcal{O}_K$ , consequently $\beta\mathcal{O}_L=A\mathcal{O}_L$ , and thus $A=\beta E$ for some unit $E\in U_L$ . This would imply the contradiction $H=A^{\sigma-1}=(\beta E)^{\sigma-1}=E^{\sigma-1}$ , since $\beta^{\sigma-1}=1$ . The ideal $\mathfrak{j}$ has yet another interesting property. The $\ell$ th power of its extension ideal $\mathfrak{j}^{\ell}\mathcal{O}_L=(\mathfrak{j}\mathcal{O}_L)^{\ell}=\mathrm{N}_{L\vert K}(A\mathcal{O}_L)=\mathrm{N}_{L\vert K}(A)\mathcal{O}_L$ coincides with its relative norm and thus, by forming the intersection with $\mathcal{O}_K$ , turns out to be principal in the base field $K$ already.

Eventually, Hilbert is in the position to state his celebrated Theorem 94: If $L\vert K$ is a cyclic extension of number fields of odd prime degree $\ell$ with trivial relative discriminant $\mathfrak{d}_{L\vert K}=\mathcal{O}_K$ , that is, an unramified extension, then there exists a non-principal ideal $\mathfrak{j}\in\mathcal{I}_K\setminus\mathcal{P}_K$ of the base field $K$ which becomes principal $\mathfrak{j}\mathcal{O}_L=A\mathcal{O}_L\in\mathcal{P}_L$ in the extension field $L$ , but the $\ell$ th power of this non-principal ideal is principal $\mathfrak{j}^{\ell}=\mathrm{N}_{L\vert K}(A)\mathcal{O}_K\in\mathcal{P}_K$ in the base field $K$ already. Hence, the class number of the base field must be divisible by $\ell$ and the extension field $L$ can be called a class field of $K$ .

Theorem 94 includes the simple inequality $\#\ker(j_{L\vert K})\ge\ell=\lbrack L:K\rbrack$ for the order of the principalization kernel of the extension $L\vert K$ . However, an improved estimate by a possibly bigger lower bound can be derived by means of the theorem on the Herbrand quotient ^[11] $h(G,U_L)$ of the $G$ -module $U_L$ , which is given by $h(G,U_L):=\#H^{-1}(G,U_L)/\#H^0(G,U_L)=(\ker(N):\mathrm{im}(\Delta))/(\ker(\Delta):\mathrm{im}(N))$ $=(E_{L\vert K}:U_L^{\sigma-1})/(U_K:\mathrm{N}_{L\vert K}(U_L))=\lbrack L:K\rbrack$ , where $L\vert K$ is a (not necessarily unramified) relative extension of odd degree $\lbrack L:K\rbrack$ (not necessarily a prime) with cyclic Galois group $G=\mathrm{Gal}(L\vert K)$ . With the aid of K. Iwasawa's isomorphism ^[12] between the first cohomology group $H^1(G,U_L)$ of $G$ in $U_L$ and the quotient $\mathcal{P}^G_L/\mathcal{P}_K$ of the group of ambiguous principal ideals of $L$ modulo the group of principal ideals of $K$ , for any Galois extension $L\vert K$ with automorphism group $G=\mathrm{Gal}(L\vert K)$ , specialized to a cyclic extension with periodic cohomology of length $2$ , and observing that $\mathcal{P}^G_L=\mathcal{P}_L\cap\mathcal{I}_K$ consists of extension ideals only when $L\vert K$ is unramified, we obtain $\#\ker(j_{L\vert K})=\#(\mathcal{P}_L\cap\mathcal{I}_K/\mathcal{P}_K)=\#(\mathcal{P}^G_L/\mathcal{P}_K)=\#H^1(G,U_L)=\#H^{-1}(G,U_L)$ $=\lbrack L:K\rbrack\cdot\#H^0(G,U_L)=\lbrack L:K\rbrack\cdot (U_K:\mathrm{N}_{L\vert K}(U_L))$ . This relation increases the lower bound by the factor $(U_K:\mathrm{N}_{L\vert K}(U_L))$ , the so-called unit norm index.

History

As mentioned in the lead section, several investigators tried to generalize the Hilbert-Artin-Furtwängler principal ideal theorem of 1930 to questions concerning the principalization in intermediate extensions between the base field and its Hilbert cass field. On the one hand, they established general theorems on the principalization over arbitrary number fields, such as Ph. Furtwängler 1932, ^[13] O. Taussky 1932, ^[14] O. Taussky 1970, ^[15] and H. Kisilevsky 1970. ^[16] On the other hand, they searched for concrete numerical examples of principalization in unramified cyclic extensions of particular kinds of base fields.

Quadratic fields

The principalization of $3$ -classes of complex quadratic number fields $K=\mathbb{Q}(\sqrt{d})$ with $3$ -class rank two in unramified cyclic cubic extensions was calculated manually for three discriminants $d\in\lbrace -3299,-4027,-9748\rbrace$ by A. Scholz and O. Taussky ^[7] in 1934. Since these calculations require composition of binary quadratic forms and explicit knowledge of fundamental systems of units in cubic number fields, which was a very difficult task in 1934, the investigations stayed at rest for half a century until F.-P. Heider and B. Schmithals ^[17] employed the CDC Cyber 76 computer at the University of Cologne to extend the information concerning principalization to the range $-2\cdot 10^4 < d < 10^5$ containing $27$ relevant discriminants in 1982, thereby providing the first analysis of five real quadratic fields. Two years later, J. R. Brink ^[18] computed the principalization types of $66$ complex quadratic fields. Currently, the most extensive computation of principalization data for all $4596$ quadratic fields with discriminants $-10^6 < d < 10^7$ and $3$ -class group of type $(3,3)$ is due to D. C. Mayer in 2010, ^[19] who used his recently discovered connection between transfer kernels and transfer targets for the design of a new principalization algorithm. ^[20]

The $2$ -principalization in unramified quadratic extensions of complex quadratic fields with $2$ -class group of type $(2,2)$ was studied by H. Kisilevsky in 1976. ^[21] Similar investigations of real quadratic fields were carried out by E. Benjamin and C. Snyder in 1995. ^[22]

Cubic fields

The $2$ -principalization in unramified quadratic extensions of cyclic cubic number fields with $2$ -class group of type $(2,2)$ was investigated by A. Derhem in 1988. ^[23] Seven years later, M. Ayadi studied the $3$ -principalization in unramified cyclic cubic extensions of cyclic cubic fields $K\subset\mathbb{Q}(\zeta_f)$ , $\zeta_f^f=1$ , with $3$ -class group of type $(3,3)$ and conductor $f$ divisible by two or three primes. ^[24]

Sextic fields

In 1992, M. C. Ismaili investigated the $3$ -principalization in unramified cyclic cubic extensions of the normal closure of pure cubic fields $K=\mathbb{Q}(\sqrt[3]{D})$ , in the case that this sextic number field $N=K(\zeta_3)$ , $\zeta_3^3=1$ , has a $3$ -class group of type $(3,3)$ . ^[25]

Quartic fields

In 1993, A. Azizi studied the $2$ -principalization in unramified quadratic extensions of bicyclic biquadratic fields of Dirichlet type $K=\mathbb{Q}(\sqrt{d},\sqrt{-1})$ with $2$ -class group of type $(2,2)$ . ^[26] Most recently, in 2014, A. Zekhnini extended the investigations to Dirichlet fields with $2$ -class group of type $(2,2,2)$ , ^[27] thus providing the first examples of $2$ -principalization in the two layers of unramified quadratic and biquadratic extensions of quartic fields with class groups of $2$ -rank three.

Secondary sources

Cassels, J.W.S.; Fröhlich, Albrecht, eds. (1967). Algebraic Number Theory. Academic Press. Zbl 0153.07403.
Iwasawa, Kenkichi (1986). Local class field theory. Oxford Mathematical Monographs. Oxford University Press. ISBN 978-0-19-504030-2. MR 863740. Zbl 0604.12014.
Janusz, Gerald J. (1973). Algebraic number fields. Pure and Applied Mathematics 55. Academic Press. p. 142. Zbl 0307.12001.
Neukirch, Jürgen (1999). Algebraic Number Theory. Grundlehren der Mathematischen Wissenschaften 322. Springer-Verlag. ISBN 978-3-540-65399-8. MR 1697859. Zbl 0956.11021.
Neukirch, Jürgen; Schmidt, Alexander; Wingberg, Kay (2008). Cohomology of Number Fields. Grundlehren der Mathematischen Wissenschaften 323 (2nd ed.). Springer-Verlag. ISBN 3-540-37888-X. Zbl 1136.11001.

References

↑ Hurwitz, A. (1926). "Über Beziehungen zwischen den Primidealen eines algebraischen Körpers und den Substitutionen seiner Gruppe". Math. Z. 25: 661–665. doi:10.1007/bf01283860.
↑ 2.0 2.1 Hilbert, D. (1897). "Die Theorie der algebraischen Zahlkörper". Jahresber. Deutsch. Math. Verein. 4: 175–546.
↑ 3.0 3.1 Hasse, H. (1930). "Bericht über neuere Untersuchungen und Probleme aus der Theorie der algebraischen Zahlkörper. Teil II: Reziprozitätsgesetz". Jahresber. Deutsch. Math. Verein., Ergänzungsband 6: 1–204.
↑ Artin, E. (1927). "Beweis des allgemeinen Reziprozitätsgesetzes". Abh. Math. Sem. Univ. Hamburg 5: 353–363.
↑ 5.0 5.1 Miyake, K. (1989). "Algebraic investigations of Hilbert's Theorem 94, the principal ideal theorem and the capitulation problem". Expo. Math. 7: 289–346.
↑ Artin, E. (1929). "Idealklassen in Oberkörpern und allgemeines Reziprozitätsgesetz". Abh. Math. Sem. Univ. Hamburg 7: 46–51.
↑ 7.0 7.1 Scholz, A., Taussky, O. (1934). "Die Hauptideale der kubischen Klassenkörper imaginär quadratischer Zahlkörper: ihre rechnerische Bestimmung und ihr Einfluß auf den Klassenkörperturm". J. Reine Angew. Math. 171: 19–41.
↑ Schreier, O. (1926). "Über die Erweiterung von Gruppen II". Abh. Math. Sem. Univ. Hamburg 4: 321–346.
↑ Newman, M. F. (1977). Determination of groups of prime-power order. pp. 73-84, in: Group Theory, Canberra, 1975, Lecture Notes in Math., Vol. 573, Springer, Berlin.
↑ O'Brien, E. A. (1990). "The p-group generation algorithm". J. Symbolic Comput. 9: 677–698. doi:10.1016/s0747-7171(08)80082-x.
↑ Herbrand, J. (1932). "Sur les théorèmes du genre principal et des idéaux principaux". Abh. Math. Sem. Univ. Hamburg 9: 84–92. doi:10.1007/bf02940630.
↑ Iwasawa, K. (1956). "A note on the group of units of an algebraic number field". J. Math. Pures Appl. 9 (35): 189–192.
↑ Furtwängler, Ph. (1932). "Über eine Verschärfung des Hauptidealsatzes für algebraische Zahlkörper". J. Reine Angew. Math. 167: 379–387.
↑ Taussky, O. (1932). "Über eine Verschärfung des Hauptidealsatzes für algebraische Zahlkörper". J. Reine Angew. Math. 168: 193–210.
↑ Taussky, O. (1970). "A remark concerning Hilbert's Theorem 94". J. Reine Angew. Math. 239/240: 435–438.
↑ Kisilevsky, H. (1970). "Some results related to Hilbert's Theorem 94". J. Number Theory 2: 199–206. doi:10.1016/0022-314x(70)90020-x.
↑ Heider, F.-P., Schmithals, B. (1982). "Zur Kapitulation der Idealklassen in unverzweigten primzyklischen Erweiterungen". J. Reine Angew. Math. 363: 1–25.
↑ Brink, J. R. (1984). The class field tower for imaginary quadratic number fields of type (3,3). Dissertation, Ohio State Univ.
↑ Mayer, D. C. (2012). "The second p-class group of a number field". Int. J. Number Theory 8 (2): 471–505. doi:10.1142/s179304211250025x.
↑ Mayer, D. C. (2014). "Principalization algorithm via class group structure". J. Théor. Nombres Bordeaux 26 (2): 415–464.
↑ Kisilevsky, H. (1976). "Number fields with class number congruent to 4 mod 8 and Hilbert's Theorem 94". J. Number Theory 8: 271–279. doi:10.1016/0022-314x(76)90004-4.
↑ Benjamin, E., Snyder, C. (1995). "Real quadratic number fields with 2-class group of type (2,2)". Math. Scand. 76: 161–178.
↑ Derhem, A. (1988). Capitulation dans les extensions quadratiques non ramifiées de corps de nombres cubiques cycliques. Thèse de Doctorat, Univ. Laval, Québec.
↑ Ayadi, M. (1995). Sur la capitulation de 3-classes d'idéaux d'un corps cubique cyclique. Thèse de Doctorat, Univ. Laval, Québec.
↑ Ismaili, M. C. (1992). Sur la capitulation de 3-classes d'idéaux de la clôture normale d'un corps cubique pure. Thèse de Doctorat, Univ. Laval, Québec.
↑ Azizi, A. (1993). Sur la capitulation de 2-classes d'idéaux de $\mathbb{Q}(\sqrt{d},i)$ . Thèse de Doctorat, Univ. Laval, Québec.
↑ Zekhnini, A. (2014). Capitulation des 2-classes d'idéaux de certains corps de nombres biquadratiques imaginaires $\mathbb{Q}(\sqrt{d},i)$ de type (2,2,2). Thèse de Doctorat, Univ. Mohammed Premier, Faculté des Sciences d'Oujda, Maroc.
↑ Jaulent, J.-F. (26 February 1988). "L'état actuel du problème de la capitulation". Séminaire de Théorie des Nombres de Bordeaux 17: 1–33.