Principalization (algebra) explained

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

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

$$\backslash begin\backslash iota\_:\; \backslash mathcal\_K\backslash to\backslash mathcal\_L\backslash \backslash \; \backslash mathfrak\backslash mapsto\backslash mathfrak\backslash mathcal\_L\; \backslash end$$

is an injective group homomorphism. Since

\iota _{L/K} ( l{P} _{K)\subseteql{P}} _{L} , this map induces the extension homomorphism of ideal class groups

$$\backslash beginj\_:\; \backslash mathcal\_K/\backslash mathcal\_K\backslash to\backslash mathcal\_L/\backslash mathcal\_L\; \backslash \backslash \; \backslash mathfrak\backslash mathcal\_K\; \backslash mapsto\; (\backslash mathfrak\backslash mathcal\_L)\; \backslash mathcal\_L\; \backslash end$$

If there exists a non-principal ideal

ak{a}\in l{I} _{K} (i.e. ak{a} l{P} _{K} \ne l{P} _{K} ) whose extension ideal in L is principal (i.e. ak{a} l{O} _{L} = A l{O} _{L} for some A\in l{O} _{L} and ( ak{a} l{O} _{L)l{P}} _{L=(Al{O}} _{L)l{P}} _{L=l{P}} _{L} ), then we speak about principalization or capitulation in L/K . In this case, the ideal ak{a} and its class ak{a} l{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/K} ) of the class extension homomorphism.

More generally, let

ak{m}= ak{m} _{0ak{m}} _{infty} be a modulus in K , where ak{m} _{0} is a nonzero ideal in l{O} _{K} and ak{m} _{infty} is a formal product of pair-wise different real infinite primes of K . Then

$$\backslash mathcal\_\; =\backslash langle\backslash alpha\backslash mathcal\_K\; |\; \backslash alpha\backslash equiv\; 1\; \backslash bmod\; \backslash rangle\; \backslash le\; \backslash mathcal\_K\; (\backslash mathfrak),$$

is the ray modulo

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

$$\backslash begin\; j\_:\; \backslash mathcal\_K(\backslash mathfrak\_K)/\backslash mathcal\_K\backslash to\backslash mathcal\_L(\backslash mathfrak\_L)/\backslash mathcal\_L\; \backslash \backslash \; \backslash mathfrak\backslash mathcal\_K\backslash mapsto(\backslash mathfrak\backslash mathcal\_L)\backslash mathcal\_L\; \backslash end$$

Galois extensions of number fields

Let

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

Frobenius automorphism

There exists a unique automorphism

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

$$I\_=\backslash \; =\backslash ker(D\_\; \backslash to\backslash mathrm(\backslash mathcal\_F/\backslash mathfrak|\backslash mathcal\_K/\backslash mathfrak))$$

e( ak{P}| ak{p}) of ak{P} over ak{p} ). Any other prime ideal of F dividing ak{p} is of the form \tau( ak{P}) with some \tau\in G . Its Frobenius automorphism is given by

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

since

$$\backslash tau(A)^\backslash equiv(\backslash tau\backslash sigma\backslash tau^)(\backslash tau(A))\backslash bmod$$

for all

A\in l{O} _{F} , and thus its decomposition group D _{\tau(ak{P} )} = \tau D _{ak{P} }\tau^ is conjugate to D _{ak{P} }. In this general situation, the Artin symbol is a mapping

$$\backslash mathfrak\backslash mapsto\backslash left(\backslash frac\backslash right):=\backslash left.\; \backslash left\backslash$$

which associates an entire conjugacy class of automorphisms to any unramified prime ideal

ak{p}\nmid ak{d} , and we have $\backslash left(\backslash frac\backslash right)=1$ if and only if ak{p} splits completely in F .

Factorization of prime ideals

When

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

We have

$$\backslash \#(H\backslash cdot\backslash tau\_i\; D\_)\backslash cdot\backslash \#D\_=\backslash \#H\backslash tau\_iD\_=\backslash \#(H\backslash tau\_i\backslash cdot\; D\_)\backslash cdot\backslash \#H.$$

Let

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

$$f(\backslash mathfrak\_i|\backslash mathfrak)\; =\; \backslash frac=\backslash frac=\backslash frac=\backslash frac=\backslash frac=\backslash \#(H\backslash tau\_i\backslash cdot\; D\_).$$

In other words, the inertia degree

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

Consequently, the ideal embedding is given by $\backslash iota\_(\backslash mathfrak)=\backslash mathfrak\backslash mathcal\_L\; =\backslash prod\_^g\backslash mathfrak\_i$, and the class extension by

$$j\_(\backslash mathfrak\backslash mathcal\_K)=(\backslash mathfrak\backslash mathcal\_L)\backslash mathcal\_L=\backslash prod\_^g\; \backslash mathfrak\_i\backslash mathcal\_L.$$

Artin's reciprocity law

Now further assume

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

By class field theory,^[3] the abelian extension

F/K uniquely corresponds to an intermediate group l{S} _{K,ak{f} } \le\mathcal \le \mathcal_K(\mathfrak) between the ray modulo ak{f} of K and l{I} _{K(ak{f})} , where ak{f}= ak{f} _{0ak{f}} _{infty=ak{f}(F/K)} denotes the relative conductor (ak{f} _{0} is divisible by the same prime ideals as ak{d} ). The Artin symbol

$$\backslash begin\; \backslash mathbb\_K(\backslash mathfrak)\backslash to\; G\backslash \backslash \; \backslash mathfrak\backslash mapsto\backslash left(\backslash frac\backslash right)\backslash end$$

which associates the Frobenius automorphism of

ak{p} to each prime ideal ak{p} of K which is unramified in F , can be extended by multiplicativity to a surjective homomorphism

$$\backslash begin\; \backslash mathcal\_K(\backslash mathfrak)\backslash to\; G\backslash \backslash \; \backslash mathfrak=\backslash prod\; \backslash mathfrak^\backslash mapsto\backslash left(\backslash frac\backslash right):=\backslash prod\; \backslash left(\backslash frac\backslash right)^\backslash end$$

with kernel

l{H}= l{S} _{K,ak{f} }\cdot\mathrm_(\mathcal_F(\mathfrak)) (where l{I} _{F(ak{f})} means l{I} _{F(ak{f}} _{0l{O}} _{F)} ), called Artin map, which induces isomorphism

$$\backslash begin\backslash mathcal\_K(\backslash mathfrak)/\backslash mathcal\backslash to\; G=\backslash mathrm(F/K)\backslash \backslash \; \backslash mathfrak\backslash mathcal\backslash mapsto\backslash left(\backslash frac\; \backslash right)\; \backslash end$$

of the generalized ideal class group

l{I} _{K(ak{f})/l{H}} to the Galois group 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

K\subseteq L\subseteq F based on the following scenario from number theory to group theory. Let F/K be a Galois extension of algebraic number fields with automorphism group G = Gal (F/K) . Assume that K\subseteq L\subseteq F is an intermediate field with relative group H = Gal (F/L)\le G and let K'/K, L'/L be the maximal abelian subextension of K, L respectively within F . Then the corresponding relative groups are the commutator subgroups G' = Gal (F/K')\le G , resp. H' = Gal (F/L')\le H . By class field theory, there exist intermediate groups l{S} _{K,ak{m} _{K}} \le l{H} _{K\le} l{I} _{K(ak{d})} and l{S} _{L,ak{m} _{L}\lel{H}} _{L\lel{I}} _{L(ak{d})} such that the Artin maps establish isomorphisms

Here

ak{d}= ak{d}(F/K), l{I} _{L(ak{d})} means l{I} _{L(ak{d}l{O}} _{L)} and ak{m} _{K,ak{m}} _{L} are some moduli divisible by ak{f}(K'/K), ak{f}(L'/L) respectively and by all primes dividing ak{d}, ak{d} l{O} _{L} respectively.

The ideal extension homomorphism

\iota _{L/K} : l{I} _{K(ak{d})\tol{I}} _{L(ak{d})} , the induced Artin transfer \tilde{T} _{G,H} and these Artin maps are connected by the formula

$$\backslash tilde\_\backslash circ\backslash left(\backslash frac\backslash right)=\backslash left(\backslash frac\backslash right)\backslash circ\; \backslash iota\_.$$

Since

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

$$\backslash prod\_^g\backslash left(\backslash frac\backslash right)=\; \backslash prod\_^g\; \backslash left[\backslash frac\{F/L\}\{\backslash tau\_i(\backslash mathfrak\{P\})\}\backslash right]\backslash cdot\; H\text{'}=\backslash prod\_^g\; \backslash tau\_i\backslash left[\backslash frac\{F/L\}\{\backslash mathfrak\{P\}\}\backslash right]\backslash tau\_i^\backslash cdot\; H\text{'}=\; \backslash prod\_^g\; \backslash tau\_i\backslash left[\backslash frac\{F/K\}\{\backslash mathfrak\{P\}\}\; \backslash right]^\; \backslash tau\_i^\backslash cdot\; H\text{'},$$

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 $\backslash left(\backslash frac\backslash right)$ of the base field

K maps the ideal ak{p} to the Frobenius automorphism $\backslash left(\backslash frac\backslash right)=\backslash left[\backslash frac\{F/K\}\{\backslash mathfrak\{P\}\}\backslash right]\backslash cdot\; G\text{'}$. The g -tuple (\tau ^{-1} _{1} ,\ldots,\tau ^{-1} _{g} ) is a system of representatives of double cosets D _{ak{P} }\backslash G/H, which correspond to the orbits of the action of $\backslash left[\backslash frac\{F/K\}\{\backslash mathfrak\{P\}\}\backslash right]$ on the set of left cosets G/H by left multiplication, and f _{i=\#(H\tau} _{i ⋅} D _{ak{P} })=\#(D_\cdot\tau_i^H) is equal to the size of the orbit of coset \tau ^{-1} _{i} H in this action. Hence the induced Artin transfer maps $\backslash left[\backslash frac\{F/K\}\{\backslash mathfrak\{P\}\}\backslash right]\backslash cdot\; G\text{'}$ to the product

$$\backslash tilde\_\backslash left(\backslash left[\backslash frac\{F/K\}\{\backslash mathfrak\{P\}\}\backslash right]\backslash cdot\; G\text{'}\backslash right)=\; T\_\backslash left(\backslash left[\backslash frac\{F/K\}\{\backslash mathfrak\{P\}\}\; \backslash right]\; \backslash right)=\; \backslash prod\_^g\; (\backslash tau\_i^)^\backslash left[\backslash frac\{F/K\}\{\backslash mathfrak\{P\}\}\backslash right]^\backslash tau\_i^\backslash cdot\; H\text{'}=\; \backslash prod\_^g\; \backslash tau\_i\; \backslash left[\backslash frac\{F/K\}\{\backslash mathfrak\{P\}\}\backslash right]^\backslash tau\_i^\backslash cdot\; H\text{'}.$$

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

\left(\tfrac{K'/K}{ ⋅ }\right) and \left(\tfrac{L'/L}{ ⋅ }\right) are l{H} _{K} and l{H} _{L} respectively, the previous formula implies that \iota _{L/K} ( l{H} _{K)\subseteql{H}} _{L} . It follows that there is the class extension homomorphism j _{L/K} : l{I} _{K(ak{d})/l{H}} _{K\tol{I}} _{L(ak{d})/l{H}} _{L} and that j _{L/K} and the induced Artin transfer \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 \tilde{T} _{G,H} \circ\left(\tfrac{K'/K}{ ⋅ }\right) = \left( \tfrac{L'/L}{ ⋅ } \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

j _{L/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 F = F ^{2(K)} is the second Hilbert class field of K , that is the maximal metabelian unramified extension of K (and maximal abelian unramified extension of F ^{1(K)} ). Then K' = L, L' = F, ak{d}= l{O} _{K,} l{H} _{K=l{P}} _{K,} l{H} _{L} = l{P} _{L} and H = G' is the commutator subgroup of G . More precisely, Furtwängler showed that generally the Artin transfer T _{G,G'} from a finite metabelian group G to its derived subgroup G' is a trivial homomorphism. In fact this is true even if G isn't metabelian because we can reduce to the metabelian case by replacing G with G/G'' . It also holds for infinite groups provided G is finitely generated and [G:G']< inf ty . It follows that every ideal of K extends to a principal ideal of F ^{1(K)} .

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 = Gal (F ^{2} _{p(K)/L)\le} G = Gal (F ^{2} _{p(K)/K)} correspondingly varies over the intermediate groups between G and G' , computation of all principalization kernels \ker(j _{L/K} ) and all p-class groups Cl _{p(L)} translates to information on the kernels \ker(T _{G,H} ) and targets H/H' of the Artin transfers T _{G,H} and permits the exact specification of the second p-class group G = Gal (F ^{2} _{p(K)/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 Gal (F ^{inf ty} _{p(K)/K)} of the maximal unramified pro-p extension F ^{inf ty} _{p(K)} of K .

These ideas are explicit in the paper of 1934 by A. Scholz and O. Taussky already. 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.^[7] Nowadays, we use the p-group generation algorithm of M. F. Newman^[8] and E. A. O'Brien^[9] 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

L/K of algebraic number fields with cyclic Galois group G = Gal (L/K) = \langle\sigma\rangle generated by an automorphism \sigma such that \sigma ^{\ell=1} for the relative degree \ell = [ L:K] , 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

$$\backslash begin\; \backslash Delta:\; U\backslash to\; U\; \backslash \backslash \; E\backslash mapsto\; E^:=\backslash sigma(E)/E\; \backslash end$$

is the symbolic exponentiation with the difference

\sigma -1 \in\Z[ G] , and the second endomorphism

$$\backslash begin\; N:\; U\backslash to\; U\; \backslash \backslash \; E\backslash mapsto\; E^:=\backslash prod\_^\backslash sigma^i(E)\; \backslash end$$

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

$$T\_G=\backslash sum\_^\backslash sigma^i\backslash in\backslash Z[G].$$

In fact, the image of the algebraic norm map is contained in the unit group

U _{K} of the base field and N(E) = 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 \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 Tate cohomology group of

G in U _{L} is given by the quotient H ^{0 (G,U} _{L): = \ker(\Delta)/ im (N) =} U _{K/N} _{L/K} (U _{L)} consisting of the norm residues of U _{K} , and the minus first Tate cohomology group of G in U _{L} is given by the quotient H ^{-1} (G,U _{L):=\ker(N)/im(\Delta)=E} _{L/K} /U ^{\sigma -1} _{L} of the group E _{L/K} = \{ E\in U _{L|} N(E) =1 \} of relative units of L/K modulo the subgroup of symbolic powers of units with formal exponent \sigma -1 .

In his Theorem 92 Hilbert proves the existence of a relative unit

H\in E _{L/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/K} /U ^{\sigma -1} _{L} 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 ^{x} ) = \{ A\in L ^{x} | N(A) =1 \}/ (L ^{x} ) ^{\sigma-1} of the G -module L ^{x} = L\setminus\{ 0 \} , the multiplicative group of the superfield L , can be defined, and Hilbert shows its triviality H ^{-1} (G,L ^{x} ) =1 in his famous Theorem 90.

Eventually, Hilbert is in the position to state his celebrated Theorem 94: If

L/K is a cyclic extension of number fields of odd prime degree \ell with trivial relative discriminant ak{d} _{L/K} = l{O} _{K} , which means it's unramified at finite primes, then there exists a non-principal ideal ak{j}\in l{I} _{K\setminusl{P}} _{K} of the base field K which becomes principal in the extension field L , that is ak{j} l{O} _{L=Al{O}} _{L\inl{P}} _{L} for some A\in l{O} _{L} . Furthermore, the \ell th power of this non-principal ideal is principal in the base field K , in particular ak{j} ^{\ell} = N _{L/K} (A) l{O} _{K\inl{P}} _{K} , 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 . The proof goes as follows: Theorem 92 says there exists unit H\in E _{L/K} \setminus U ^{\sigma -1} _{L} , then Theorem 90 ensures the existence of a (necessarily non-unit) A\in L ^{x} such that H = A ^{\sigma-1} , i. e., A ^{\sigma} = A ⋅ H . By multiplying A by proper integer if necessary we may assume that A is an algebraic integer. The non-unit A is generator of an ambiguous principal ideal of L/K , since (A l{O} ^{\sigma} _{L)} = A ^{\sigma} l{O} _{L=A ⋅} H l{O} _{L=Al{O}} _{L} . However, the underlying ideal ak{j}:=(A l{O} _{L)\capl{O}} _{K} of the subfield K cannot be principal. Assume to the contrary that ak{j}=\beta l{O} _{K} for some \beta\in l{O} _{K} . Since L/K is unramified, every ambiguous ideal ak{a} of l{O} _{L} is a lift of some ideal in l{O} _{K} , in particular ak{a}=( ak{a}\cap l{O} _{K)l{O}} _{L} . Hence \beta l{O} _{L=ak{j}l{O}} _{L=Al{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} because \beta ^{\sigma-1} =1 . On the other hand,

$$\backslash mathfrak^\backslash mathcal\_L=\; (\backslash mathfrak\backslash mathcal\_L)^\; =\backslash mathrm\_\; (\backslash mathfrak\backslash mathcal\_L)\; \backslash mathcal\_L=\; \backslash mathrm\_(A\backslash mathcal\_L)\backslash mathcal\_L=\backslash mathrm\_(A)\backslash mathcal\_L,$$

thus

ak{j} ^{\ell} = N _{L/K} (A) l{O} _{K} is principal in the base field K already.

Theorems 92 and 94 don't hold as stated for

\ell =2 , with the fields K = \Q(\sqrt{ 3 }) and L = K(i) being a counterexample (in this particular case L is the narrow Hilbert class field of K ). The reason is Hilbert only considers ramification at finite primes but not at infinite primes (we say that a real infinite prime of K ramifies in L if there exists non-real extension of this prime to L ). This doesn't make a difference when [L:K] is odd since the extension is then unramified at infinite primes. However he notes that Theorems 92 and 94 hold for \ell =2 provided we further assume that number of fields conjugate to L that are real is twice the number of real fields conjugate to K . This condition is equivalent to L/K being unramified at infinite primes, so Theorem 94 holds for all primes \ell if we assume that L/K is unramified everywhere.

Theorem 94 implies the simple inequality

\#\ker(j _{L/K} )\ge\ell = [L:K] for the order of the principalization kernel of the extension L/K . However an exact formula for the order of this kernel can be derived for cyclic unramified (including infinite primes) extension (not necessarily of prime degree) by means of the Herbrand quotient^[10] h(G,U _{L)} of the G -module U _{L} , which is given by

$$h(G,U\_L):=\backslash \#H^(G,U\_L)/\backslash \#H^0(G,U\_L)=(\backslash ker(N):\backslash mathrm(\backslash Delta))/(\backslash ker(\backslash Delta):\backslash mathrm(N))=(E\_:U\_L^)/(U\_K:\backslash mathrm\_(U\_L)).$$

It can be shown that

h(G,U _{L)=[} L:K] (without calculating the order of either of the cohomology groups). Since the extension L/K is unramified, it's l{I} ^{G} _{L = l{I}} _{Kl{O}} _{L} so l{P} ^{G} _{L} = l{P} _{L} \cap l{I} _{Kl{O}} _{L} . With the aid of K. Iwasawa's isomorphism^[11] H ^{G} _{L/ l{P}} _{Kl{O}} _{L} , specialized to a cyclic extension with periodic cohomology of length 2 , we obtain

This relation increases the lower bound by the factor

(U _{K:N} _{L/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 class field. On the one hand, they established general theorems on the principalization over arbitrary number fields, such as Ph. Furtwängler 1932,^[12] O. Taussky 1932,^[13] O. Taussky 1970,^[14] and H. Kisilevsky 1970.^[15] 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 imaginary quadratic fields K = \Q(\sqrt{d}) with 3 -class rank two in unramified cyclic cubic extensions was calculated manually for three discriminants d\in\{ -3299 , -4027 , -9748 \} by A. Scholz and O. Taussky^[16] 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 ⋅ 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 imaginary 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 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\Q(\zeta _{f)} , \zeta ^{f =1} _{f} , 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 = \Q(\sqrt[ 3 ]{D}) , in the case that this sextic number field N = K(\zeta _{3)} , \zeta ^{3=1} _{3} , 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 biquadratic fields of Dirichlet type K = \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.

See also

Both, the algebraic, group theoretic access to the principalization problem by Hilbert-Artin-Furtwängler and the arithmetic, cohomological access by Hilbert-Herbrand-Iwasawa are also presented in detail in the two bibles of capitulation by J.-F. Jaulent 1988^[28] and by K. Miyake 1989.^[6]

Secondary sources

• Book: J.W.S. . Cassels . J. W. S. Cassels . Albrecht . Fröhlich . Albrecht Fröhlich . Algebraic Number Theory . 1967 . Academic Press . 0153.07403 .
• Book: Kenkichi Iwasawa

. Iwasawa . Kenkichi . Kenkichi Iwasawa . Local class field theory . Oxford University Press . Oxford Mathematical Monographs . 978-0-19-504030-2 . 863740 . 1986 . 0604.12014 .

• Book: Janusz, Gerald J. . Algebraic number fields . Pure and Applied Mathematics . 55 . Academic Press . 1973 . 142 . 0307.12001 .
• Book: Neukirch, Jürgen . Jürgen Neukirch

. Jürgen Neukirch . Algebraic Number Theory . 322 . Grundlehren der Mathematischen Wissenschaften . . 1999 . 978-3-540-65399-8 . 0956.11021 . 1697859 .

• Book: Cohomology of Number Fields . 323 . Grundlehren der Mathematischen Wissenschaften . de . Jürgen . Neukirch . Jürgen Neukirch . Alexander . Schmidt . Kay . Wingberg . 2nd . . 2008 . 978-3-540-37888-4 . 1136.11001 .

Notes and References

1. Hurwitz, A.. 1926. Über Beziehungen zwischen den Primidealen eines algebraischen Körpers und den Substitutionen seiner Gruppe. de. Math. Z.. 25. 661–665 . 10.1007/bf01283860. 119971823.
2. Hilbert, D.. 1897. Die Theorie der algebraischen Zahlkörper. de. Jahresber. Deutsch. Math. Verein.. 4. 175–546.
3. Hasse, H.. 1930. Bericht über neuere Untersuchungen und Probleme aus der Theorie der algebraischen Zahlkörper. Teil II: Reziprozitätsgesetz. de. Jahresber. Deutsch. Math. Verein., Ergänzungsband. 6. 1–204.
4. Artin, E.. 1927. Beweis des allgemeinen Reziprozitätsgesetzes. de. Abh. Math. Sem. Univ. Hamburg. 5. 353–363. 10.1007/BF02952531. 123050778.
5. Artin, E.. 1929. Idealklassen in Oberkörpern und allgemeines Reziprozitätsgesetz. de. Abh. Math. Sem. Univ. Hamburg. 7. 46–51. 10.1007/BF02941159. 121475651.
6. Miyake, K.. 1989. Algebraic investigations of Hilbert's Theorem 94, the principal ideal theorem and the capitulation problem. Expo. Math.. 7. 289–346.
7. Schreier, O.. 1926. Über die Erweiterung von Gruppen II. de. Abh. Math. Sem. Univ. Hamburg. 4. 321–346. 10.1007/BF02950735. 122947636.
8. Book: 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.
9. O'Brien, E. A.. 1990. The p-group generation algorithm. J. Symbolic Comput.. 9. 5–6. 677–698. 10.1016/s0747-7171(08)80082-x. free.
10. Herbrand, J.. 1932. Sur les théorèmes du genre principal et des idéaux principaux . fr . Abh. Math. Sem. Univ. Hamburg. 9. 84–92. 10.1007/bf02940630. 120775483.
11. Iwasawa, K.. 1956. A note on the group of units of an algebraic number field. J. Math. Pures Appl.. 9. 35. 189–192.
12. Furtwängler, Ph.. 1932. Über eine Verschärfung des Hauptidealsatzes für algebraische Zahlkörper. de. J. Reine Angew. Math.. 1932. 167. 379–387. 10.1515/crll.1932.167.379. 199546266.
13. Taussky, O.. 1932. Über eine Verschärfung des Hauptidealsatzes für algebraische Zahlkörper. de. J. Reine Angew. Math.. 1932. 168. 193–210. 10.1515/crll.1932.168.193. 199545623.
14. Taussky, O.. 1970. A remark concerning Hilbert's Theorem 94. J. Reine Angew. Math.. 239/240. 435–438.
15. Kisilevsky, H.. 1970. Some results related to Hilbert's Theorem 94. J. Number Theory. 2. 2. 199–206. 10.1016/0022-314x(70)90020-x. 1970JNT.....2..199K. free.
16. Scholz, A., Taussky, O.. 1934. Die Hauptideale der kubischen Klassenkörper imaginär quadratischer Zahlkörper: ihre rechnerische Bestimmung und ihr Einfluß auf den Klassenkörperturm. de. J. Reine Angew. Math.. 171. 19–41.
17. Heider, F.-P., Schmithals, B.. 1982. Zur Kapitulation der Idealklassen in unverzweigten primzyklischen Erweiterungen. de. J. Reine Angew. Math.. 363. 1–25.
18. Book: Brink, J. R.. 1984. The class field tower for imaginary quadratic number fields of type (3,3). Dissertation, Ohio State Univ..
19. Mayer, D. C.. 2012. The second p-class group of a number field. Int. J. Number Theory. 8. 2. 471–505. 10.1142/s179304211250025x. 1403.3899. 119332361.
20. Mayer, D. C.. 2014. Principalization algorithm via class group structure. J. Théor. Nombres Bordeaux. 26. 2. 415–464. 10.5802/jtnb.874. 1403.3839. 119740132.
21. Kisilevsky, H.. 1976. Number fields with class number congruent to 4 mod 8 and Hilbert's Theorem 94. J. Number Theory. 8. 3. 271–279. 10.1016/0022-314x(76)90004-4. free.
22. Benjamin, E., Snyder, C.. 1995. Real quadratic number fields with 2-class group of type (2,2). Math. Scand.. 76. 161–178. 10.7146/math.scand.a-12532. free.
23. Book: Derhem, A.. 1988. Capitulation dans les extensions quadratiques non ramifiées de corps de nombres cubiques cycliques. Thèse de Doctorat, Univ. Laval, Québec. fr.
24. Book: Ayadi, M.. 1995. Sur la capitulation de 3-classes d'idéaux d'un corps cubique cyclique. fr. Thèse de Doctorat, Univ. Laval, Québec.
25. Book: Ismaili, M. C.. 1992. Sur la capitulation de 3-classes d'idéaux de la clôture normale d'un corps cubique pure. fr. Thèse de Doctorat, Univ. Laval, Québec.
26. Book: Azizi, A.. 1993. Sur la capitulation de 2-classes d'idéaux de \Q(\sqrt{d},i) . fr . Thèse de Doctorat, Univ. Laval, Québec.
27. Book: Zekhnini, A.. 2014. Capitulation des 2-classes d'idéaux de certains corps de nombres biquadratiques imaginaires \Q(\sqrt{d},i) de type (2,2,2) . fr . Thèse de Doctorat, Univ. Mohammed Premier, Faculté des Sciences d'Oujda, Maroc.
28. Jaulent, J.-F.. 26 February 1988. L'état actuel du problème de la capitulation. fr. Séminaire de Théorie des Nombres de Bordeaux. 17. 1–33.