Let p be a prime and K be a number field with non-trivial p-class group ClpK. A crucial step in identifying the Galois group G∞p of the maximal unramified pro-p extension of K is to determine its two-stage approximat...Let p be a prime and K be a number field with non-trivial p-class group ClpK. A crucial step in identifying the Galois group G∞p of the maximal unramified pro-p extension of K is to determine its two-stage approximation M=G2pk, that is the second derived quotient M≃G/Gn. The family τ1K of abelian type invariants of the p-class groups ClpL of all unramified cyclic extensions L/K of degree p is called the index- abelianization data (IPAD) of K. It is able to specify a finite batch of contestants for the second p-class group M of K. In this paper we introduce two different kinds of generalized IPADs for obtaining more sophisticated results. The multi-layered IPAD (τ1Kτ(2)K) includes data on unramified abelian extensions L/K of degree p2 and enables sharper bounds for the order of M in the case Clpk≃(p,p,p), where current im-plementations of the p-group generation algorithm fail to produce explicit contestants for M , due to memory limitations. The iterated IPAD of second order τ(2)K contains information on non-abelian unramified extensions L/K of degree p2, or even p3, and admits the identification of the p-class tower group G for various infinite series of quadratic fields K=Q(√d) with ClpK≃(p,p) possessing a p-class field tower of exact length lpK=3 as a striking novelty.展开更多
Theoretical foundations of a new algorithm for determining the p-capitulation type ù(K) of a number field K with p-class rank ?=2 are presented. Since ù(K) alone is insufficient for identifying the seco...Theoretical foundations of a new algorithm for determining the p-capitulation type ù(K) of a number field K with p-class rank ?=2 are presented. Since ù(K) alone is insufficient for identifying the second p-class group G=Gal(F<sub>p</sub><sup>2</sup>K∣K) of K, complementary techniques are deve- loped for finding the nilpotency class and coclass of . An implementation of the complete algorithm in the computational algebra system Magma is employed for calculating the Artin pattern AP(K)=(τ (K),ù(K)) of all 34631 real quadratic fields K=Q(√d) with discriminants 0d<10<sup>8</sup> and 3-class group of type (3, 3). The results admit extensive statistics of the second 3-class groups G=Gal(F<sub>3</sub><sup>2</sup>K∣K) and the 3-class field tower groups G=Gal(F<sub>3</sub><sup>∞</sup>K∣K).展开更多
We focus on orders in arbitrary number fields, consider their Picard groups and finally obtain ring class fields corresponding to them. The Galois group of the ring class field is isomorphic to the Picard group.As an ...We focus on orders in arbitrary number fields, consider their Picard groups and finally obtain ring class fields corresponding to them. The Galois group of the ring class field is isomorphic to the Picard group.As an application, we give criteria of the integral solvability of the diophantine equation p = x2+ ny2 over a class of imaginary quadratic fields where p is a prime element.展开更多
Recent examples of periodic bifurcations in descendant trees of finite p-groups with ?are used to show that the possible p-class tower groups G of certain multiquadratic fields K with p- class group of type (2,2,2) , ...Recent examples of periodic bifurcations in descendant trees of finite p-groups with ?are used to show that the possible p-class tower groups G of certain multiquadratic fields K with p- class group of type (2,2,2) , resp. (3,3), form periodic sequences in the descendant tree of the elementary Abelian root , resp. . The particular vertex of the periodic sequence which occurs as the p-class tower group G of an assigned field K is determined uniquely by the p-class number of a quadratic, resp. cubic, auxiliary field k, associated unambiguously to K. Consequently, the hard problem of identifying the p-class tower group G is reduced to an easy computation of low degree arithmetical invariants.展开更多
Given a fixed prime number p, the multiplet of abelian type invariants of the p-class groups of all unramified cyclic degree p extensions of a number field K is called its IPAD (index-p abeliani- zation data). These i...Given a fixed prime number p, the multiplet of abelian type invariants of the p-class groups of all unramified cyclic degree p extensions of a number field K is called its IPAD (index-p abeliani- zation data). These invariants have proved to be a valuable information for determining the Galois group of the second Hilbert p-class field and the p-capitulation type of K. For p=3 and a number field K with elementary p-class group of rank two, all possible IPADs are given in the complete form of several infinite sequences. Iterated IPADs of second order are used to identify the group of the maximal unramified pro-p extension of K.展开更多
Let p be a prime. For any finite p-group G, the deep transfers T H,G ' : H / H ' → G ' / G " from the maximal subgroups H of index (G:H) = p in G to the derived subgroup G ' are introduced as an ...Let p be a prime. For any finite p-group G, the deep transfers T H,G ' : H / H ' → G ' / G " from the maximal subgroups H of index (G:H) = p in G to the derived subgroup G ' are introduced as an innovative tool for identifying G uniquely by means of the family of kernels ùd(G) =(ker(T H,G ')) (G: H) = p. For all finite 3-groups G of coclass cc(G) = 1, the family ùd(G) is determined explicitly. The results are applied to the Galois groups G =Gal(F3 (∞)/ F) of the Hilbert 3-class towers of all real quadratic fields F = Q(√d) with fundamental discriminants d > 1, 3-class group Cl3(F) □ C3 × C3, and total 3-principalization in each of their four unramified cyclic cubic extensions E/F. A systematic statistical evaluation is given for the complete range 1 d 7, and a few exceptional cases are pointed out for 1 d 8.展开更多
We present the entanglement measures of a tetrapartite W-class entangled system in a noninertial frame, where the transformation between Minkowski and Rindler coordinates is applied.Two cases are considered.First, whe...We present the entanglement measures of a tetrapartite W-class entangled system in a noninertial frame, where the transformation between Minkowski and Rindler coordinates is applied.Two cases are considered.First, when one qubit has uniform acceleration whilst the other three remain stationary.Second, when two qubits have nonuniform accelerations and the others stay inertial.The 1–1 tangle, 1–3 tangle, and whole entanglement measurements π4 and Π4, are studied and illustrated with graphics through their dependence on the acceleration parameter rd for the first case and rc and rd for the second case.It is found that the negativities(1–1 tangle and 1–3 tangle) and π-tangle decrease when the acceleration parameter rd or in the second case rc and rd increase, remaining a nonzero entanglement in the majority of the results.This means that the system will be always entangled except for special cases.It is shown that only the 1–1 tangle for the first case vanishes at infinite accelerations, but for the second case the 1–1 tangle disappears completely when r > 0.472473.An analytical expression for the von Neumann information entropy of the system is found and we note that it increases with the acceleration parameter.展开更多
The Lipschitz class Lip(K, α) on a local field K is defined in [10], and an equivalent relationship between the Ho¨lder type space Cα(K)[9] and Lip(K,α) is given. In this note, we give a 'chain of function...The Lipschitz class Lip(K, α) on a local field K is defined in [10], and an equivalent relationship between the Ho¨lder type space Cα(K)[9] and Lip(K,α) is given. In this note, we give a 'chain of function spaces' over Euclidian space by defining higher order continuous modulus in R, and point out that there is no need of higher order continuous modulus for describing the chain of function spaces over local fields.展开更多
New methanol maser lines at 72 → 63A^-（86.6 GHz） and 72 → 63A^＋（86.9 GHz） together with two candidate methanol maser lines at 72 → 81A-（80.99GHz） and 72→81A^＋（111.29GHz） have been detected in W3（OH）. W...New methanol maser lines at 72 → 63A^-（86.6 GHz） and 72 → 63A^＋（86.9 GHz） together with two candidate methanol maser lines at 72 → 81A-（80.99GHz） and 72→81A^＋（111.29GHz） have been detected in W3（OH）. We use a pumping mechanism, i.e., methanol masers without population inversion, to explain the formation of weak methanol masers of 72 → 81A^＋ and 72→ 81A^-. We explain well why the line-shape of the transition 72 → 81A^＋ is not typical. A similar argument can be applied to the A-type level system 72A^-, 63A^- and 81A^-, as well as to the 72 → 81A^- 80.99 GHz masers.展开更多
By making use of our generalization of Barrucand and Cohn’s theory of principal factorizations in pure cubic fields and their Galois closures with 3 possible types to pure quintic fields and their pure metacyclic nor...By making use of our generalization of Barrucand and Cohn’s theory of principal factorizations in pure cubic fields and their Galois closures with 3 possible types to pure quintic fields and their pure metacyclic normal fields with 13 possible types, we compile an extensive database with arithmetical invariants of the 900 pairwise non-isomorphic fields N having normalized radicands in the range 2≤D3. Our classification is based on the Galois cohomology of the unit group UN, viewed as a module over the automorphism group Gal(N/K) of N over the cyclotomic field K=Q(ξ5), by employing theorems of Hasse and Iwasawa on the Herbrand quotient of the unit norm index (Uk:NN/K(UN)) by the number #(PN/K/PK) of primitive ambiguous principal ideals, which can be interpreted as principal factors of the different DN/K. The precise structure of the F5-vector space of differential principal factors is expressed in terms of norm kernels and central orthogonal idempotents. A connection with integral representation theory is established via class number relations by Parry and Walter involving the index of subfield units (UN:U0).?The statistical distribution of the 13 principal factorization types and their refined splitting into similarity classes with representative prototypes is discussed thoroughly.展开更多
Let F be a number field and p be a prime. In the successive approximation theorem, we prove that, for each integer n ≥ 1, finitely many candidates for the Galois group of the nth stage of the p-class tower over F are...Let F be a number field and p be a prime. In the successive approximation theorem, we prove that, for each integer n ≥ 1, finitely many candidates for the Galois group of the nth stage of the p-class tower over F are determined by abelian type invariants of p-class groups C1pE of unramified extensions E/F with degree [E : F] = pn-1. Illustrated by the most extensive numerical results available currently, the transfer kernels (TE, F) of the p-class extensions TE, F : C1pF → C1pE from F to unramified cyclic degree-p extensions E/F are shown to be capable of narrowing down the number of contestants significantly. By determining the isomorphism type of the maximal subgroups S G of all 3-groups G with coclass cc(G) = 1, and establishing a general theorem on the connection between the p-class towers of a number field F and of an unramified abelian p-extension E/F, we are able to provide a theoretical proof of the realization of certain 3-groups S with maximal class by 3-tower groups of dihedral fields E with degree 6, which could not be realized up to now.展开更多
文摘Let p be a prime and K be a number field with non-trivial p-class group ClpK. A crucial step in identifying the Galois group G∞p of the maximal unramified pro-p extension of K is to determine its two-stage approximation M=G2pk, that is the second derived quotient M≃G/Gn. The family τ1K of abelian type invariants of the p-class groups ClpL of all unramified cyclic extensions L/K of degree p is called the index- abelianization data (IPAD) of K. It is able to specify a finite batch of contestants for the second p-class group M of K. In this paper we introduce two different kinds of generalized IPADs for obtaining more sophisticated results. The multi-layered IPAD (τ1Kτ(2)K) includes data on unramified abelian extensions L/K of degree p2 and enables sharper bounds for the order of M in the case Clpk≃(p,p,p), where current im-plementations of the p-group generation algorithm fail to produce explicit contestants for M , due to memory limitations. The iterated IPAD of second order τ(2)K contains information on non-abelian unramified extensions L/K of degree p2, or even p3, and admits the identification of the p-class tower group G for various infinite series of quadratic fields K=Q(√d) with ClpK≃(p,p) possessing a p-class field tower of exact length lpK=3 as a striking novelty.
文摘Theoretical foundations of a new algorithm for determining the p-capitulation type ù(K) of a number field K with p-class rank ?=2 are presented. Since ù(K) alone is insufficient for identifying the second p-class group G=Gal(F<sub>p</sub><sup>2</sup>K∣K) of K, complementary techniques are deve- loped for finding the nilpotency class and coclass of . An implementation of the complete algorithm in the computational algebra system Magma is employed for calculating the Artin pattern AP(K)=(τ (K),ù(K)) of all 34631 real quadratic fields K=Q(√d) with discriminants 0d<10<sup>8</sup> and 3-class group of type (3, 3). The results admit extensive statistics of the second 3-class groups G=Gal(F<sub>3</sub><sup>2</sup>K∣K) and the 3-class field tower groups G=Gal(F<sub>3</sub><sup>∞</sup>K∣K).
基金supported by National Natural Science Foundation of China(Grant No.11471314)the National Basic Research Program of China(973 Project)(Grant No.2011CB302401)the National Center for Mathematics and Interdisciplinary Sciences,Chinese Academy of Sciences
文摘We focus on orders in arbitrary number fields, consider their Picard groups and finally obtain ring class fields corresponding to them. The Galois group of the ring class field is isomorphic to the Picard group.As an application, we give criteria of the integral solvability of the diophantine equation p = x2+ ny2 over a class of imaginary quadratic fields where p is a prime element.
基金supported by the NSFC(1080110530871444)+2 种基金the Key Project of Natural Science of Anhui Provincial Department of Education(KJ2009A44)the Doctoral Special Fund of Hefei University of Technology(GDBJ2010-012)the Key Project of Science and Technology of Anhui Province(08010302070)
文摘In this paper, we give a lower bound exp（2.2 × 10~8 ） for those discriminants of real quadratic fields Q（√ d） with d= N^2-4 and h（d）=1.
文摘Recent examples of periodic bifurcations in descendant trees of finite p-groups with ?are used to show that the possible p-class tower groups G of certain multiquadratic fields K with p- class group of type (2,2,2) , resp. (3,3), form periodic sequences in the descendant tree of the elementary Abelian root , resp. . The particular vertex of the periodic sequence which occurs as the p-class tower group G of an assigned field K is determined uniquely by the p-class number of a quadratic, resp. cubic, auxiliary field k, associated unambiguously to K. Consequently, the hard problem of identifying the p-class tower group G is reduced to an easy computation of low degree arithmetical invariants.
文摘Given a fixed prime number p, the multiplet of abelian type invariants of the p-class groups of all unramified cyclic degree p extensions of a number field K is called its IPAD (index-p abeliani- zation data). These invariants have proved to be a valuable information for determining the Galois group of the second Hilbert p-class field and the p-capitulation type of K. For p=3 and a number field K with elementary p-class group of rank two, all possible IPADs are given in the complete form of several infinite sequences. Iterated IPADs of second order are used to identify the group of the maximal unramified pro-p extension of K.
文摘Let p be a prime. For any finite p-group G, the deep transfers T H,G ' : H / H ' → G ' / G " from the maximal subgroups H of index (G:H) = p in G to the derived subgroup G ' are introduced as an innovative tool for identifying G uniquely by means of the family of kernels ùd(G) =(ker(T H,G ')) (G: H) = p. For all finite 3-groups G of coclass cc(G) = 1, the family ùd(G) is determined explicitly. The results are applied to the Galois groups G =Gal(F3 (∞)/ F) of the Hilbert 3-class towers of all real quadratic fields F = Q(√d) with fundamental discriminants d > 1, 3-class group Cl3(F) □ C3 × C3, and total 3-principalization in each of their four unramified cyclic cubic extensions E/F. A systematic statistical evaluation is given for the complete range 1 d 7, and a few exceptional cases are pointed out for 1 d 8.
基金Project partially supported by the CONACYT,Mexico under the Grant No.288856-CB-2016partially by 20190234-SIP-IPN,Mexicopartially by the program XXVIII Verano de la Investigación Científica 2018 supported by the Academia Mexicana de Ciencias
文摘We present the entanglement measures of a tetrapartite W-class entangled system in a noninertial frame, where the transformation between Minkowski and Rindler coordinates is applied.Two cases are considered.First, when one qubit has uniform acceleration whilst the other three remain stationary.Second, when two qubits have nonuniform accelerations and the others stay inertial.The 1–1 tangle, 1–3 tangle, and whole entanglement measurements π4 and Π4, are studied and illustrated with graphics through their dependence on the acceleration parameter rd for the first case and rc and rd for the second case.It is found that the negativities(1–1 tangle and 1–3 tangle) and π-tangle decrease when the acceleration parameter rd or in the second case rc and rd increase, remaining a nonzero entanglement in the majority of the results.This means that the system will be always entangled except for special cases.It is shown that only the 1–1 tangle for the first case vanishes at infinite accelerations, but for the second case the 1–1 tangle disappears completely when r > 0.472473.An analytical expression for the von Neumann information entropy of the system is found and we note that it increases with the acceleration parameter.
文摘The Lipschitz class Lip(K, α) on a local field K is defined in [10], and an equivalent relationship between the Ho¨lder type space Cα(K)[9] and Lip(K,α) is given. In this note, we give a 'chain of function spaces' over Euclidian space by defining higher order continuous modulus in R, and point out that there is no need of higher order continuous modulus for describing the chain of function spaces over local fields.
基金Supported by the National Natural Science Foundation of China.
文摘New methanol maser lines at 72 → 63A^-（86.6 GHz） and 72 → 63A^＋（86.9 GHz） together with two candidate methanol maser lines at 72 → 81A-（80.99GHz） and 72→81A^＋（111.29GHz） have been detected in W3（OH）. We use a pumping mechanism, i.e., methanol masers without population inversion, to explain the formation of weak methanol masers of 72 → 81A^＋ and 72→ 81A^-. We explain well why the line-shape of the transition 72 → 81A^＋ is not typical. A similar argument can be applied to the A-type level system 72A^-, 63A^- and 81A^-, as well as to the 72 → 81A^- 80.99 GHz masers.
文摘By making use of our generalization of Barrucand and Cohn’s theory of principal factorizations in pure cubic fields and their Galois closures with 3 possible types to pure quintic fields and their pure metacyclic normal fields with 13 possible types, we compile an extensive database with arithmetical invariants of the 900 pairwise non-isomorphic fields N having normalized radicands in the range 2≤D3. Our classification is based on the Galois cohomology of the unit group UN, viewed as a module over the automorphism group Gal(N/K) of N over the cyclotomic field K=Q(ξ5), by employing theorems of Hasse and Iwasawa on the Herbrand quotient of the unit norm index (Uk:NN/K(UN)) by the number #(PN/K/PK) of primitive ambiguous principal ideals, which can be interpreted as principal factors of the different DN/K. The precise structure of the F5-vector space of differential principal factors is expressed in terms of norm kernels and central orthogonal idempotents. A connection with integral representation theory is established via class number relations by Parry and Walter involving the index of subfield units (UN:U0).?The statistical distribution of the 13 principal factorization types and their refined splitting into similarity classes with representative prototypes is discussed thoroughly.
文摘Let F be a number field and p be a prime. In the successive approximation theorem, we prove that, for each integer n ≥ 1, finitely many candidates for the Galois group of the nth stage of the p-class tower over F are determined by abelian type invariants of p-class groups C1pE of unramified extensions E/F with degree [E : F] = pn-1. Illustrated by the most extensive numerical results available currently, the transfer kernels (TE, F) of the p-class extensions TE, F : C1pF → C1pE from F to unramified cyclic degree-p extensions E/F are shown to be capable of narrowing down the number of contestants significantly. By determining the isomorphism type of the maximal subgroups S G of all 3-groups G with coclass cc(G) = 1, and establishing a general theorem on the connection between the p-class towers of a number field F and of an unramified abelian p-extension E/F, we are able to provide a theoretical proof of the realization of certain 3-groups S with maximal class by 3-tower groups of dihedral fields E with degree 6, which could not be realized up to now.
