Principalization (algebra)

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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

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Let   be an algebraic number field, called the base field, and let   be a field extension of finite degree. Let   and   denote the ring of integers, the group of nonzero fractional ideals and its subgroup of principal fractional ideals of the fields   respectively. Then the extension map of fractional ideals

 

is an injective group homomorphism. Since  , this map induces the extension homomorphism of ideal class groups

 

If there exists a non-principal ideal   (i.e.  ) whose extension ideal in   is principal (i.e.   for some   and  ), then we speak about principalization or capitulation in  . In this case, the ideal   and its class   are said to principalize or capitulate in  . This phenomenon is described most conveniently by the principalization kernel or capitulation kernel, that is the kernel   of the class extension homomorphism.

More generally, let   be a modulus in  , where   is a nonzero ideal in   and   is a formal product of pair-wise different real infinite primes of  . Then

 

is the ray modulo  , where   is the group of nonzero fractional ideals in   relatively prime to   and the condition   means   and   for every real infinite prime   dividing   Let   then the group   is called a generalized ideal class group for   If   and   are generalized ideal class groups such that   for every   and   for every  , then   induces the extension homomorphism of generalized ideal class groups:

 

Galois extensions of number fields

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Let   be a Galois extension of algebraic number fields with Galois group   and let   denote the set of prime ideals of the fields   respectively. Suppose that   is a prime ideal of   which does not divide the relative discriminant  , and is therefore unramified in  , and let   be a prime ideal of   lying over  .

Frobenius automorphism

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There exists a unique automorphism   such that   for all algebraic integers  , where   is the norm of  . The map   is called the Frobenius automorphism of  . It generates the decomposition group   of   and its order is equal to the inertia degree   of   over  . (If   is ramified then   is only defined and generates   modulo the inertia subgroup

 

whose order is the ramification index   of   over  ). Any other prime ideal of   dividing   is of the form   with some  . Its Frobenius automorphism is given by

 

since

 

for all  , and thus its decomposition group   is conjugate to  . In this general situation, the Artin symbol is a mapping

 

which associates an entire conjugacy class of automorphisms to any unramified prime ideal  , and we have   if and only if   splits completely in  .

Factorization of prime ideals

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When   is an intermediate field with relative Galois group  , more precise statements about the homomorphisms   and   are possible because we can construct the factorization of   (where   is unramified in   as above) in   from its factorization in   as follows.[1][2] Prime ideals in   lying over   are in  -equivariant bijection with the  -set of left cosets  , where   corresponds to the coset  . For every prime ideal   in   lying over   the Galois group   acts transitively on the set of prime ideals in   lying over  , thus such ideals   are in bijection with the orbits of the action of   on   by left multiplication. Such orbits are in turn in bijection with the double cosets  . Let   be a complete system of representatives of these double cosets, thus  . Furthermore, let   denote the orbit of the coset   in the action of   on the set of left cosets   by left multiplication and let   denote the orbit of the coset   in the action of   on the set of right cosets   by right multiplication. Then   factorizes in   as  , where   for   are the prime ideals lying over   in   satisfying   with the product running over any system of representatives of  .

We have

 

Let   be the decomposition group of   over  . Then   is the stabilizer of   in the action of   on  , so by the orbit-stabilizer theorem we have  . On the other hand, it's  , which together gives

 

In other words, the inertia degree   is equal to the size of the orbit of the coset   in the action of   on the set of right cosets   by right multiplication. By taking inverses, this is equal to the size of the orbit   of the coset   in the action of   on the set of left cosets   by left multiplication. Also the prime ideals in   lying over   correspond to the orbits of this action.

Consequently, the ideal embedding is given by  , and the class extension by

 

Artin's reciprocity law

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Now further assume   is an abelian extension, that is,   is an abelian group. Then, all conjugate decomposition groups of prime ideals of   lying over   coincide, thus   for every  , and the Artin symbol   becomes equal to the Frobenius automorphism of any   and   for all   and every  .

By class field theory,[3] the abelian extension   uniquely corresponds to an intermediate group   between the ray modulo   of   and  , where   denotes the relative conductor (  is divisible by the same prime ideals as  ). The Artin symbol

 

which associates the Frobenius automorphism of   to each prime ideal   of   which is unramified in  , can be extended by multiplicativity to a surjective homomorphism

 

with kernel   (where   means  ), called Artin map, which induces isomorphism

 

of the generalized ideal class group   to the Galois group  . 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.

Group-theoretic formulation of the problem

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This reciprocity law allowed Artin to translate the general principalization problem for number fields   based on the following scenario from number theory to group theory. Let   be a Galois extension of algebraic number fields with automorphism group  . Assume that   is an intermediate field with relative group   and let   be the maximal abelian subextension of   respectively within  . Then the corresponding relative groups are the commutator subgroups  , resp.  . By class field theory, there exist intermediate groups   and   such that the Artin maps establish isomorphisms

 

Here   means   and   are some moduli divisible by   respectively and by all primes dividing   respectively.

The ideal extension homomorphism  , the induced Artin transfer   and these Artin maps are connected by the formula

 

Since   is generated by the prime ideals of   which does not divide  , it's enough to verify this equality on these generators. Hence suppose that   is a prime ideal of   which does not divide   and let   be a prime ideal of   lying over  . On the one hand, the ideal extension homomorphism   maps the ideal   of the base field   to the extension ideal   in the field  , and the Artin map   of the field   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   of the base field   maps the ideal   to the Frobenius automorphism  . The  -tuple   is a system of representatives of double cosets  , which correspond to the orbits of the action of   on the set of left cosets   by left multiplication, and   is equal to the size of the orbit of coset   in this action. Hence the induced Artin transfer maps   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.[5]

Since the kernels of the Artin maps   and   are   and   respectively, the previous formula implies that  . It follows that there is the class extension homomorphism   and that   and the induced Artin transfer   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  .[3][6]

Class field tower

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The commutative diagram in the previous section, which connects the number theoretic class extension homomorphism   with the group theoretic Artin transfer  , enabled Furtwängler to prove the principal ideal theorem by specializing to the situation that   is the (first) Hilbert class field of  , that is the maximal abelian unramified extension of  , and   is the second Hilbert class field of  , that is the maximal metabelian unramified extension of   (and maximal abelian unramified extension of  ). Then   and   is the commutator subgroup of  . More precisely, Furtwängler showed that generally the Artin transfer   from a finite metabelian group   to its derived subgroup   is a trivial homomorphism. In fact this is true even if   isn't metabelian because we can reduce to the metabelian case by replacing   with  . It also holds for infinite groups provided   is finitely generated and  . It follows that every ideal of   extends to a principal ideal of  .

However, the commutative diagram comprises the potential for a lot of more sophisticated applications. In the situation that   is a prime number,   is the second Hilbert p-class field of  , that is the maximal metabelian unramified extension of   of degree a power of   varies over the intermediate field between   and its first Hilbert p-class field  , and   correspondingly varies over the intermediate groups between   and  , computation of all principalization kernels   and all p-class groups   translates to information on the kernels   and targets   of the Artin transfers   and permits the exact specification of the second p-class group   of   via pattern recognition, and frequently even allows to draw conclusions about the entire p-class field tower of  , that is the Galois group   of the maximal unramified pro-p extension   of  .

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

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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   of algebraic number fields with cyclic Galois group   generated by an automorphism   such that   for the relative degree  , which is assumed to be an odd prime.

He investigates two endomorphism of the unit group   of the extension field, viewed as a Galois module with respect to the group  , briefly a  -module. The first endomorphism

 

is the symbolic exponentiation with the difference  , 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   of the base field and   coincides with the usual arithmetic (field) norm as the product of all conjugates. The composita of the endomorphisms satisfy the relations   and  .

Two important cohomology groups can be defined by means of the kernels and images of these endomorphisms. The zeroth Tate cohomology group of   in   is given by the quotient   consisting of the norm residues of  , and the minus first Tate cohomology group of   in   is given by the quotient   of the group   of relative units of   modulo the subgroup of symbolic powers of units with formal exponent  .

In his Theorem 92 Hilbert proves the existence of a relative unit   which cannot be expressed as  , for any unit  , which means that the minus first cohomology group   is non-trivial of order divisible by  . However, with the aid of a completely similar construction, the minus first cohomology group   of the  -module  , the multiplicative group of the superfield  , can be defined, and Hilbert shows its triviality   in his famous Theorem 90.

Eventually, Hilbert is in the position to state his celebrated Theorem 94: If   is a cyclic extension of number fields of odd prime degree   with trivial relative discriminant  , which means it's unramified at finite primes, then there exists a non-principal ideal   of the base field   which becomes principal in the extension field  , that is   for some  . Furthermore, the  th power of this non-principal ideal is principal in the base field  , in particular  , hence the class number of the base field must be divisible by   and the extension field   can be called a class field of  . The proof goes as follows: Theorem 92 says there exists unit  , then Theorem 90 ensures the existence of a (necessarily non-unit)   such that  , i. e.,  . By multiplying   by proper integer if necessary we may assume that   is an algebraic integer. The non-unit   is generator of an ambiguous principal ideal of  , since  . However, the underlying ideal   of the subfield   cannot be principal. Assume to the contrary that   for some  . Since   is unramified, every ambiguous ideal   of   is a lift of some ideal in  , in particular  . Hence   and thus   for some unit  . This would imply the contradiction   because  . On the other hand,

 

thus   is principal in the base field   already.

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

Theorem 94 implies the simple inequality   for the order of the principalization kernel of the extension  . 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[11]   of the  -module  , which is given by

 

It can be shown that   (without calculating the order of either of the cohomology groups). Since the extension   is unramified, it's   so   . With the aid of K. Iwasawa's isomorphism[12]  , specialized to a cyclic extension with periodic cohomology of length  , we obtain

 

This relation increases the lower bound by the factor  , the so-called unit norm index.

History

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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,[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

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The principalization of  -classes of imaginary quadratic fields   with  -class rank two in unramified cyclic cubic extensions was calculated manually for three discriminants   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   containing   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   complex quadratic fields. Currently, the most extensive computation of principalization data for all   quadratic fields with discriminants   and  -class group of type   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  -principalization in unramified quadratic extensions of imaginary quadratic fields with  -class group of type   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

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The  -principalization in unramified quadratic extensions of cyclic cubic fields with  -class group of type   was investigated by A. Derhem in 1988.[23] Seven years later, M. Ayadi studied the  -principalization in unramified cyclic cubic extensions of cyclic cubic fields  ,  , with  -class group of type   and conductor   divisible by two or three primes.[24]

Sextic fields

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In 1992, M. C. Ismaili investigated the  -principalization in unramified cyclic cubic extensions of the normal closure of pure cubic fields  , in the case that this sextic number field  ,  , has a  -class group of type  .[25]

Quartic fields

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In 1993, A. Azizi studied the  -principalization in unramified quadratic extensions of biquadratic fields of Dirichlet type   with  -class group of type  .[26] Most recently, in 2014, A. Zekhnini extended the investigations to Dirichlet fields with  -class group of type  ,[27] thus providing the first examples of  -principalization in the two layers of unramified quadratic and biquadratic extensions of quartic fields with class groups of  -rank three.

See also

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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

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  • 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 0863740. Zbl 0604.12014.
  • Janusz, Gerald J. (1973). Algebraic number fields. Pure and Applied Mathematics. Vol. 55. Academic Press. p. 142. Zbl 0307.12001.
  • Neukirch, Jürgen (1999). Algebraic Number Theory. Grundlehren der Mathematischen Wissenschaften. Vol. 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 (in German). Vol. 323 (2nd ed.). Springer-Verlag. ISBN 978-3-540-37888-4. Zbl 1136.11001.

References

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  1. ^ Hurwitz, A. (1926). "Über Beziehungen zwischen den Primidealen eines algebraischen Körpers und den Substitutionen seiner Gruppe". Math. Z. (in German). 25: 661–665. doi:10.1007/bf01283860. S2CID 119971823.
  2. ^ a b Hilbert, D. (1897). "Die Theorie der algebraischen Zahlkörper". Jahresber. Deutsch. Math. Verein. (in German). 4: 175–546.
  3. ^ a b Hasse, H. (1930). "Bericht über neuere Untersuchungen und Probleme aus der Theorie der algebraischen Zahlkörper. Teil II: Reziprozitätsgesetz". Jahresber. Deutsch. Math. Verein., Ergänzungsband (in German). 6: 1–204.
  4. ^ Artin, E. (1927). "Beweis des allgemeinen Reziprozitätsgesetzes". Abh. Math. Sem. Univ. Hamburg (in German). 5: 353–363. doi:10.1007/BF02952531. S2CID 123050778.
  5. ^ Artin, E. (1929). "Idealklassen in Oberkörpern und allgemeines Reziprozitätsgesetz". Abh. Math. Sem. Univ. Hamburg (in German). 7: 46–51. doi:10.1007/BF02941159. S2CID 121475651.
  6. ^ a b Miyake, K. (1989). "Algebraic investigations of Hilbert's Theorem 94, the principal ideal theorem and the capitulation problem". Expo. Math. 7: 289–346.
  7. ^ a b 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. (in German). 171: 19–41.{{cite journal}}: CS1 maint: multiple names: authors list (link)
  8. ^ Schreier, O. (1926). "Über die Erweiterung von Gruppen II". Abh. Math. Sem. Univ. Hamburg (in German). 4: 321–346. doi:10.1007/BF02950735. S2CID 122947636.
  9. ^ 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.
  10. ^ O'Brien, E. A. (1990). "The p-group generation algorithm". J. Symbolic Comput. 9 (5–6): 677–698. doi:10.1016/s0747-7171(08)80082-x.
  11. ^ Herbrand, J. (1932). "Sur les théorèmes du genre principal et des idéaux principaux". Abh. Math. Sem. Univ. Hamburg (in French). 9: 84–92. doi:10.1007/bf02940630. S2CID 120775483.
  12. ^ Iwasawa, K. (1956). "A note on the group of units of an algebraic number field". J. Math. Pures Appl. 9 (35): 189–192.
  13. ^ Furtwängler, Ph. (1932). "Über eine Verschärfung des Hauptidealsatzes für algebraische Zahlkörper". J. Reine Angew. Math. (in German). 1932 (167): 379–387. doi:10.1515/crll.1932.167.379. S2CID 199546266.
  14. ^ Taussky, O. (1932). "Über eine Verschärfung des Hauptidealsatzes für algebraische Zahlkörper". J. Reine Angew. Math. (in German). 1932 (168): 193–210. doi:10.1515/crll.1932.168.193. S2CID 199545623.
  15. ^ Taussky, O. (1970). "A remark concerning Hilbert's Theorem 94". J. Reine Angew. Math. 239/240: 435–438.
  16. ^ Kisilevsky, H. (1970). "Some results related to Hilbert's Theorem 94". J. Number Theory. 2 (2): 199–206. Bibcode:1970JNT.....2..199K. doi:10.1016/0022-314x(70)90020-x.
  17. ^ Heider, F.-P., Schmithals, B. (1982). "Zur Kapitulation der Idealklassen in unverzweigten primzyklischen Erweiterungen". J. Reine Angew. Math. (in German). 363: 1–25.{{cite journal}}: CS1 maint: multiple names: authors list (link)
  18. ^ 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. arXiv:1403.3899. doi:10.1142/s179304211250025x. S2CID 119332361.
  20. ^ Mayer, D. C. (2014). "Principalization algorithm via class group structure". J. Théor. Nombres Bordeaux. 26 (2): 415–464. arXiv:1403.3839. doi:10.5802/jtnb.874. S2CID 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. doi:10.1016/0022-314x(76)90004-4.
  22. ^ Benjamin, E., Snyder, C. (1995). "Real quadratic number fields with 2-class group of type (2,2)". Math. Scand. 76: 161–178. doi:10.7146/math.scand.a-12532.{{cite journal}}: CS1 maint: multiple names: authors list (link)
  23. ^ Derhem, A. (1988). Capitulation dans les extensions quadratiques non ramifiées de corps de nombres cubiques cycliques (in French). Thèse de Doctorat, Univ. Laval, Québec.
  24. ^ Ayadi, M. (1995). Sur la capitulation de 3-classes d'idéaux d'un corps cubique cyclique (in French). Thèse de Doctorat, Univ. Laval, Québec.
  25. ^ Ismaili, M. C. (1992). Sur la capitulation de 3-classes d'idéaux de la clôture normale d'un corps cubique pure (in French). Thèse de Doctorat, Univ. Laval, Québec.
  26. ^ Azizi, A. (1993). Sur la capitulation de 2-classes d'idéaux de   (in French). Thèse de Doctorat, Univ. Laval, Québec.
  27. ^ Zekhnini, A. (2014). Capitulation des 2-classes d'idéaux de certains corps de nombres biquadratiques imaginaires   de type (2,2,2) (in French). 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". Séminaire de Théorie des Nombres de Bordeaux (in French). 17: 1–33.