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 formula_1 be an algebraic number field, called the "base field", and let formula_2 be a field extension of finite degree. Let formula_3 and formula_4 denote the ring of integers, the group of nonzero fractional ideals and its subgroup of principal fractional ideals of the fields formula_5 respectively. Then the extension map of fractional ideals 
is an injective group homomorphism. Since formula_7, this map induces the extension homomorphism of ideal class groups
If there exists a non-principal ideal formula_9 (i.e. formula_10) whose extension ideal in formula_11 is principal (i.e. formula_12 for some formula_13 and formula_14), then we speak about principalization or capitulation in formula_2. In this case, the ideal formula_16 and its class formula_17 are said to principalize or capitulate in formula_11. This phenomenon is described most conveniently by the principalization kernel or capitulation kernel, that is the kernel formula_19 of the class extension homomorphism.
More generally, let formula_20 be a modulus in formula_1, where formula_22 is a nonzero ideal in formula_23 and formula_24 is a formal product of pair-wise different real infinite primes of formula_1. Then 
is the ray modulo formula_27, where formula_28 is the group of nonzero fractional ideals in formula_1 relatively prime to formula_22 and the condition formula_31 means formula_32 and formula_33 for every real infinite prime formula_34 dividing formula_35 Let formula_36 then the group formula_37 is called a generalized ideal class group for formula_38 If formula_39 and formula_40 are generalized ideal class groups such that formula_41 for every formula_42 and formula_43 for every formula_44, then formula_45 induces the extension homomorphism of generalized ideal class groups:
Galois extensions of number fields.
Let formula_47 be a Galois extension of algebraic number fields with Galois group formula_48 and let formula_49 denote the set of prime ideals of the fields formula_50 respectively. Suppose that formula_51 is a prime ideal of formula_1 which does not divide the relative discriminant formula_53, and is therefore unramified in formula_54, and let formula_55 be a prime ideal of formula_54 lying over formula_57.
Frobenius automorphism.
There exists a unique automorphism formula_58 such that formula_59 for all algebraic integers formula_60, where formula_61 is the norm of formula_57. The map formula_63 is called the Frobenius automorphism of formula_64. It generates the decomposition group formula_65 of formula_64 and its order is equal to the inertia degree formula_67 of formula_64 over formula_57. (If formula_57 is ramified then formula_71 is only defined and generates formula_72 modulo the inertia subgroup 
whose order is the ramification index formula_74 of formula_64 over formula_57). Any other prime ideal of formula_54 dividing formula_57 is of the form formula_79 with some formula_80. Its Frobenius automorphism is given by 
since 
for all formula_60, and thus its decomposition group formula_84 is conjugate to formula_72. In this general situation, the Artin symbol is a mapping 
which associates an entire conjugacy class of automorphisms to any unramified prime ideal formula_87, and we have formula_88 if and only if formula_57 splits completely in formula_54.
Factorization of prime ideals.
When formula_91 is an intermediate field with relative Galois group formula_92, more precise statements about the homomorphisms formula_45 and formula_94 are possible because we can construct the factorization of formula_57 (where formula_57 is unramified in formula_54 as above) in formula_98 from its factorization in formula_99 as follows. Prime ideals in formula_99 lying over formula_57 are in formula_102-equivariant bijection with the formula_102-set of left cosets formula_104, where formula_79 corresponds to the coset formula_106. For every prime ideal formula_107 in formula_98 lying over formula_57 the Galois group formula_110 acts transitively on the set of prime ideals in formula_99 lying over formula_107, thus such ideals formula_107 are in bijection with the orbits of the action of formula_110 on formula_104 by left multiplication. Such orbits are in turn in bijection with the double cosets formula_116. Let formula_117 be a complete system of representatives of these double cosets, thus formula_118. Furthermore, let formula_119 denote the orbit of the coset formula_120 in the action of formula_110 on the set of left cosets formula_104 by left multiplication and let formula_123 denote the orbit of the coset formula_124 in the action of formula_72 on the set of right cosets formula_126 by right multiplication. Then formula_57 factorizes in formula_98 as formula_129, where formula_130 for formula_131 are the prime ideals lying over formula_57 in formula_11 satisfying formula_134 with the product running over any system of representatives of formula_119.
We have 
Let formula_137 be the decomposition group of formula_138 over formula_11. Then formula_140 is the stabilizer of formula_120 in the action of formula_110 on formula_104, so by the orbit-stabilizer theorem we have formula_144. On the other hand, it's formula_145, which together gives
In other words, the inertia degree formula_147 is equal to the size of the orbit of the coset formula_124 in the action of formula_71 on the set of right cosets formula_126 by right multiplication. By taking inverses, this is equal to the size of the orbit formula_151 of the coset formula_152 in the action of formula_71 on the set of left cosets formula_154 by left multiplication. Also the prime ideals in formula_98 lying over formula_57 correspond to the orbits of this action.
Consequently, the ideal embedding is given by formula_157, and the class extension by 
Artin's reciprocity law.
Now further assume formula_47 is an abelian extension, that is, formula_102 is an abelian group. Then, all conjugate decomposition groups of prime ideals of formula_54 lying over formula_57 coincide, thus formula_163 for every formula_80, and the Artin symbol formula_165 becomes equal to the Frobenius automorphism of any formula_166 and formula_167 for all formula_60 and every formula_166.
By class field theory,
the abelian extension formula_47 uniquely corresponds to an intermediate group formula_171 between the ray modulo formula_172 of formula_1 and formula_174, where formula_175 denotes the relative conductor (formula_176 is divisible by the same prime ideals as formula_177). The Artin symbol 
which associates the Frobenius automorphism of formula_57 to each prime ideal formula_57 of formula_1 which is unramified in formula_54, can be extended by multiplicativity to a surjective homomorphism 
with kernel formula_184 (where formula_185 means formula_186), called Artin map, which induces isomorphism 
of the generalized ideal class group formula_188 to the Galois group formula_102. This explicit isomorphism is called the Artin reciprocity law or "general reciprocity law".
Group-theoretic formulation of the problem.
This reciprocity law allowed Artin to translate the general principalization problem for number fields formula_91 based on the following scenario from number theory to group theory. Let formula_47 be a Galois extension of algebraic number fields with automorphism group formula_48. Assume that formula_91 is an intermediate field with relative group formula_194 and let formula_195 be the maximal abelian subextension of formula_196 respectively within formula_54. Then the corresponding relative groups are the commutator subgroups formula_198, resp. formula_199. By class field theory, there exist intermediate groups formula_200 and formula_201 such that the Artin maps establish isomorphisms
Here formula_203 means formula_204 and formula_205 are some moduli divisible by formula_206 respectively and by all primes dividing formula_207 respectively.
The ideal extension homomorphism formula_208, the induced Artin transfer formula_209 and these Artin maps are connected by the formula 
Since formula_211 is generated by the prime ideals of formula_1 which does not divide formula_177, it's enough to verify this equality on these generators. Hence suppose that formula_214 is a prime ideal of formula_1 which does not divide formula_177 and let formula_217 be a prime ideal of formula_54 lying over formula_57. On the one hand, the ideal extension homomorphism formula_45 maps the ideal formula_57 of the base field formula_1 to the extension ideal formula_223 in the field formula_11, and the Artin map formula_225 of the field formula_11 maps this product of prime ideals to the product of conjugates of Frobenius automorphisms
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 formula_228 of the base field formula_1 maps the ideal formula_57 to the Frobenius automorphism formula_231. The formula_232-tuple formula_233 is a system of representatives of double cosets formula_234, which correspond to the orbits of the action of formula_71 on the set of left cosets formula_154 by left multiplication, and formula_237 is equal to the size of the orbit of coset formula_152 in this action. Hence the induced Artin transfer maps formula_239 to the product
This product expression was the original form of the Artin transfer homomorphism, corresponding to a decomposition of the permutation representation into disjoint cycles.
Since the kernels of the Artin maps formula_241 and formula_242 are formula_243 and formula_244 respectively, the previous formula implies that formula_245. It follows that there is the class extension homomorphism formula_246 and that formula_94 and the induced Artin transfer formula_209 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 formula_249.
Class field tower.
The commutative diagram in the previous section, which connects the number theoretic class extension homomorphism formula_94 with the group theoretic Artin transfer formula_251, enabled Furtwängler to prove the principal ideal theorem by specializing to the situation that formula_252 is the (first) Hilbert class field of formula_1, that is the maximal abelian unramified extension of formula_1, and formula_255 is the "second Hilbert class field" of formula_1, that is the maximal metabelian unramified extension of formula_1 (and maximal abelian unramified extension of formula_258). Then formula_259 and formula_260 is the commutator subgroup of formula_102. More precisely, Furtwängler showed that generally the Artin transfer formula_262 from a finite metabelian group formula_102 to its derived subgroup formula_264 is a trivial homomorphism. In fact this is true even if formula_102 isn't metabelian because we can reduce to the metabelian case by replacing formula_102 with formula_267. It also holds for infinite groups provided formula_102 is finitely generated and formula_269. It follows that every ideal of formula_1 extends to a principal ideal of formula_258.
However, the commutative diagram comprises the potential for a lot of more sophisticated applications. In the situation that formula_272 is a prime number, formula_273 is the "second Hilbert p-class field" of formula_1, that is the maximal metabelian unramified extension of formula_1 of degree a power of formula_276 varies over the intermediate field between formula_1 and its first "Hilbert p-class" field formula_278, and formula_279 correspondingly varies over the intermediate groups between formula_102 and formula_264, computation of all principalization kernels formula_19 and all "p-class groups" formula_283 translates to information on the kernels formula_284 and targets formula_285 of the Artin transfers formula_251 and permits the exact specification of the "second p-class group" formula_287 of formula_1 via pattern recognition, and frequently even allows to draw conclusions about the entire "p-class field tower" of formula_1, that is the Galois group formula_290 of the maximal unramified pro-"p" extension formula_291 of formula_1.
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.
Nowadays, we use the "p"-group generation algorithm of M. F. Newman
and E. A. O'Brien
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
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 formula_2 of algebraic number fields with cyclic Galois group formula_294 generated by an automorphism formula_295 such that formula_296 for the relative degree formula_297, which is assumed to be an odd prime.
He investigates two endomorphism of the unit group formula_298 of the extension field, viewed as a Galois module with respect to the group formula_102, briefly a formula_102-module. The first endomorphism 
is the symbolic exponentiation with the difference formula_302, and the second endomorphism 
is the "algebraic norm" mapping, that is the symbolic exponentiation with the trace 
In fact, the image of the algebraic norm map is contained in the unit group formula_305 of the base field and formula_306 coincides with the usual arithmetic (field) norm as the product of all conjugates. The composita of the endomorphisms satisfy the relations formula_307 and formula_308.
Two important cohomology groups can be defined by means of the kernels and images of these endomorphisms. The zeroth Tate cohomology group of formula_102 in formula_310 is given by the quotient formula_311 consisting of the "norm residues" of formula_305, and the minus first Tate cohomology group of formula_102 in formula_310 is given by the quotient formula_315 of the group formula_316 of "relative units" of formula_2 modulo the subgroup of symbolic powers of units with formal exponent formula_318.
In his Theorem 92 Hilbert proves the existence of a relative unit formula_319 which cannot be expressed as formula_320, for any unit formula_321, which means that the minus first cohomology group formula_322 is non-trivial of order divisible by formula_323. However, with the aid of a completely similar construction, the minus first cohomology group formula_324 of the formula_102-module formula_326, the multiplicative group of the superfield formula_11, can be defined, and Hilbert shows its triviality formula_328 in his famous Theorem 90.
Eventually, Hilbert is in the position to state his celebrated Theorem 94: If formula_2 is a cyclic extension of number fields of odd prime degree formula_323 with trivial relative discriminant formula_331, which means it's unramified at finite primes, then there exists a non-principal ideal formula_332 of the base field formula_1 which becomes principal in the extension field formula_11, that is formula_335 for some formula_13. Furthermore, the formula_323th power of this non-principal ideal is principal in the base field formula_1, in particular formula_339, hence the class number of the base field must be divisible by formula_323 and the extension field formula_11 can be called a class field of formula_1. The proof goes as follows: Theorem 92 says there exists unit formula_343, then Theorem 90 ensures the existence of a (necessarily non-unit) formula_344 such that formula_345, i. e., formula_346. By multiplying formula_347 by proper integer if necessary we may assume that formula_347 is an algebraic integer. The non-unit formula_347 is generator of an "ambiguous" principal ideal of formula_2, since formula_351. However, the underlying ideal formula_352 of the subfield formula_1 cannot be principal. Assume to the contrary that formula_354 for some formula_355. Since formula_2 is unramified, every ambiguous ideal formula_16 of formula_98 is a lift of some ideal in formula_23, in particular formula_360. Hence formula_361 and thus formula_362 for some unit formula_321. This would imply the contradiction formula_364 because formula_365. On the other hand, 
thus formula_367 is principal in the base field formula_1 already.
Theorems 92 and 94 don't hold as stated for formula_369, with the fields formula_370 and formula_371 being a counterexample (in this particular case formula_11 is the narrow Hilbert class field of formula_1). The reason is Hilbert only considers ramification at finite primes but not at infinite primes (we say that a real infinite prime of formula_1 ramifies in formula_11 if there exists non-real extension of this prime to formula_11). This doesn't make a difference when formula_377 is odd since the extension is then unramified at infinite primes. However he notes that Theorems 92 and 94 hold for formula_369 provided we further assume that number of fields conjugate to formula_11 that are real is twice the number of real fields conjugate to formula_1. This condition is equivalent to formula_2 being unramified at infinite primes, so Theorem 94 holds for all primes formula_323 if we assume that formula_2 is unramified everywhere.
Theorem 94 implies the simple inequality formula_384 for the order of the principalization kernel of the extension formula_2. 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 formula_386 of the formula_102-module formula_310, which is given by
It can be shown that formula_390 (without calculating the order of either of the cohomology groups). Since the extension formula_2 is unramified, it's formula_392 so formula_393 . With the aid of K. Iwasawa's isomorphism
formula_394, specialized to a cyclic extension with periodic cohomology of length formula_395, we obtain
This relation increases the lower bound by the factor formula_397, 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,
O. Taussky 1932,
O. Taussky 1970,
and H. Kisilevsky 1970.
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 formula_398-classes of imaginary quadratic fields formula_399 with formula_398-class rank two in unramified cyclic cubic extensions was calculated manually for three discriminants formula_401 by A. Scholz and O. Taussky
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
employed the CDC Cyber 76 computer at the University of Cologne to extend the information concerning principalization to the range formula_402 containing formula_403 relevant discriminants in 1982,
thereby providing the first analysis of five real quadratic fields.
Two years later, J. R. Brink
computed the principalization types of formula_404 complex quadratic fields.
Currently, the most extensive computation of principalization data for all formula_405 quadratic fields with discriminants formula_406 and formula_398-class group of type formula_408 is due to D. C. Mayer in 2010,
who used his recently discovered connection between transfer kernels and transfer targets for the design of a new "principalization algorithm".
The formula_395-principalization in unramified quadratic extensions of imaginary quadratic fields with formula_395-class group of type formula_411 was studied by H. Kisilevsky in 1976.
Similar investigations of real quadratic fields were carried out by E. Benjamin and C. Snyder in 1995.
Cubic fields.
The formula_395-principalization in unramified quadratic extensions of cyclic cubic fields with formula_395-class group of type formula_411 was investigated by A. Derhem in 1988.
Seven years later, M. Ayadi studied the formula_398-principalization in unramified cyclic cubic extensions of cyclic cubic fields formula_416, formula_417, with formula_398-class group of type formula_408 and conductor formula_420 divisible by two or three primes.
Sextic fields.
In 1992, M. C. Ismaili investigated the formula_398-principalization in unramified cyclic cubic extensions of the normal closure of pure cubic fields formula_422, in the case that this sextic number field formula_423, formula_424, has a formula_398-class group of type formula_408.
Quartic fields.
In 1993, A. Azizi studied the formula_395-principalization in unramified quadratic extensions of biquadratic fields of "Dirichlet type" formula_428 with formula_395-class group of type formula_411. Most recently, in 2014, A. Zekhnini extended the investigations to Dirichlet fields with formula_395-class group of type formula_432, thus providing the first examples of formula_395-principalization in the two layers of unramified quadratic and biquadratic extensions of quartic fields with class groups of formula_395-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 and by K. Miyake 1989.