Let L be an abelian extension of the rationals Q whose Galois group Gal(L) is an abelian (q-group q is any prime number). The explicit law of prime decomposition in L for any prime number p, the inertia group, residue...Let L be an abelian extension of the rationals Q whose Galois group Gal(L) is an abelian (q-group q is any prime number). The explicit law of prime decomposition in L for any prime number p, the inertia group, residue class degree, and discriminant of L are given here; such fields L are classified into 4 or 8 classes according as q is odd or even with clear description of their structures. Then relative extension L/K is studied. L/K is proved to have a relative integral basis under certain simple conditions; relative discriminant D(L/K) is given explicitly; and necessary and sufficient conditions are obtained for D(L/K) to be generated by a rational square (and by a rational). In particular, it is proved that L/K has a relative integral basis and that D(L/K) is generated by a rational square if [L: K]≥x~* or x~*+1 (according as q is odd or even), where x~* is the exponent of Gal(L). These results contain many related results on similar fields in literature.展开更多
A Hauser-Ernst-type extended hyperbolic complex linear system given in our previous paper [Gao Y J 2004 Chin. Phys. 13 602] is slightly modified and used to develop a new inverse scattering method for the stationary a...A Hauser-Ernst-type extended hyperbolic complex linear system given in our previous paper [Gao Y J 2004 Chin. Phys. 13 602] is slightly modified and used to develop a new inverse scattering method for the stationary axisymmetric Einstein-Maxwell theory with multiple Abelian gauge fields. The reduction procedures in this inverse scattering method are found to be fairly simple, which makes the inverse scattering method be fine and effective in practical application. As an example, a concrete family of soliton solutions for the considered theory is obtained.展开更多
For a finite abelian tame extension L/Fq(T), we will determine its genus field and conductor in this paper, and prove the existence theorems of the relative integral bases for L over any of its subfields.
Let q be any rational prime number.An abelian q-field L is a Galois extension overQ with Galois group as an abelian q-group(i.e.an abelian group whose order is a powerof q).We here will give the classification of all ...Let q be any rational prime number.An abelian q-field L is a Galois extension overQ with Galois group as an abelian q-group(i.e.an abelian group whose order is a powerof q).We here will give the classification of all the abelian q-fields L,the explicit law ofprime decomposition of any prime number p in L,the inertia groups,residue class de-grees,and discriminants of L,developing systematically and completely the results onfields of types(2,2,…,2),(q, q,…,q),and(q^s,q^s,…,q^s)in many references obtainedby specific method in each case.展开更多
Let O_L and O_K denote the rings of integers in the algebraic number fields L and K, respectively, where K is a subfield of L. If O_L is a free O_K-module, then L/K is said to have a relative integral basis. Artin and...Let O_L and O_K denote the rings of integers in the algebraic number fields L and K, respectively, where K is a subfield of L. If O_L is a free O_K-module, then L/K is said to have a relative integral basis. Artin and Frhlich raised and studied the problem of existence of a relative integral basis for a number field L. The nroblem was treated展开更多
This paper contains two parts toward studying abelian varieties from the classification point of view.In a series of papers[Doc.Math.,21,1607–1643(2016)],[Taiwan Residents J.Math.,20(4),723–741(2016)],etc.,the curre...This paper contains two parts toward studying abelian varieties from the classification point of view.In a series of papers[Doc.Math.,21,1607–1643(2016)],[Taiwan Residents J.Math.,20(4),723–741(2016)],etc.,the current authors and T.C.Yang obtain explicit formulas for the numbers of superspecial abelian surfaces over finite fields.In this paper,we give an explicit formula for the size of the isogeny class of simple abelian surfaces with real Weil number q1/2.This establishes a key step that extends our previous explicit calculation of superspecial abelian surfaces to those of supersingular abelian surfaces.The second part is to introduce the notion of genera and idealcomplexes of abelian varieties with additional structures in a general setting.The purpose is to generalize the previous work by the second named author[Forum Math.,22(3),565–582(2010)]on abelian varieties with additional structures to similitude classes,which establishes more results on the connection between geometrically defined and arithmetically defined masses for further investigations.展开更多
In this paper we prove that for any prime power q equivalent to 3 (mod 8) there exist 4 - {q(2); k, k, k, k; lambda} supplementary difference sets (SDSs) with k = q(q - 1)/2, lambda 4k - q(2), and Hadamard matrices of...In this paper we prove that for any prime power q equivalent to 3 (mod 8) there exist 4 - {q(2); k, k, k, k; lambda} supplementary difference sets (SDSs) with k = q(q - 1)/2, lambda 4k - q(2), and Hadamard matrices of order 4q(2), and give several constructions of these SDSs. Moreover, combining the results of reference [1], we conclude that for any prime p equivalent to 3 (mod 8) and integer r greater than or equal to 1 there exists an Hadamard matrix of order 4p(2r).展开更多
A theorem of Chow concerns homomorphisms of two abelian varieties under a primary field extension base change. In this paper, we generalize Chow’s theorem to semi-abelian varieties.This contributes to different proof...A theorem of Chow concerns homomorphisms of two abelian varieties under a primary field extension base change. In this paper, we generalize Chow’s theorem to semi-abelian varieties.This contributes to different proofs of a well-known result that every algebraic torus splits over a finite separable field extension. We also obtain the best bound for the degrees of splitting fields of tori.展开更多
This paper discusses quantum mechanical schemas for describing waves with non-abelian phases, Fock spaces of annihilation-creation operators for these structures, and the Feynman recipe for obtaining descriptions of p...This paper discusses quantum mechanical schemas for describing waves with non-abelian phases, Fock spaces of annihilation-creation operators for these structures, and the Feynman recipe for obtaining descriptions of particle interactions with external fields.展开更多
Quartic unramified Abelian extension fields of a class of cubic cyclic fields are given and the Hilbert class field of a cubic cyclic field with discriminant 607~2 iS obtained.
基金Project supported by the National Natural Science Foundation of China (Grant No. 19771052).
文摘Let L be an abelian extension of the rationals Q whose Galois group Gal(L) is an abelian (q-group q is any prime number). The explicit law of prime decomposition in L for any prime number p, the inertia group, residue class degree, and discriminant of L are given here; such fields L are classified into 4 or 8 classes according as q is odd or even with clear description of their structures. Then relative extension L/K is studied. L/K is proved to have a relative integral basis under certain simple conditions; relative discriminant D(L/K) is given explicitly; and necessary and sufficient conditions are obtained for D(L/K) to be generated by a rational square (and by a rational). In particular, it is proved that L/K has a relative integral basis and that D(L/K) is generated by a rational square if [L: K]≥x~* or x~*+1 (according as q is odd or even), where x~* is the exponent of Gal(L). These results contain many related results on similar fields in literature.
基金Project supported by the National Natural Science Foundation of China (Grant No 10475036)
文摘A Hauser-Ernst-type extended hyperbolic complex linear system given in our previous paper [Gao Y J 2004 Chin. Phys. 13 602] is slightly modified and used to develop a new inverse scattering method for the stationary axisymmetric Einstein-Maxwell theory with multiple Abelian gauge fields. The reduction procedures in this inverse scattering method are found to be fairly simple, which makes the inverse scattering method be fine and effective in practical application. As an example, a concrete family of soliton solutions for the considered theory is obtained.
文摘For a finite abelian tame extension L/Fq(T), we will determine its genus field and conductor in this paper, and prove the existence theorems of the relative integral bases for L over any of its subfields.
基金Project supported by the National Natural Science Foundation of China.
文摘Let q be any rational prime number.An abelian q-field L is a Galois extension overQ with Galois group as an abelian q-group(i.e.an abelian group whose order is a powerof q).We here will give the classification of all the abelian q-fields L,the explicit law ofprime decomposition of any prime number p in L,the inertia groups,residue class de-grees,and discriminants of L,developing systematically and completely the results onfields of types(2,2,…,2),(q, q,…,q),and(q^s,q^s,…,q^s)in many references obtainedby specific method in each case.
基金Project supported by the National Natural Science Foundation of China.
文摘Let O_L and O_K denote the rings of integers in the algebraic number fields L and K, respectively, where K is a subfield of L. If O_L is a free O_K-module, then L/K is said to have a relative integral basis. Artin and Frhlich raised and studied the problem of existence of a relative integral basis for a number field L. The nroblem was treated
基金the Natural Science Foundation of China(Grant No.11601395)supported by the MoST(Grant Nos.104-2115-M-001-001MY3 and 107-2115-M-001-001-MY2)。
文摘This paper contains two parts toward studying abelian varieties from the classification point of view.In a series of papers[Doc.Math.,21,1607–1643(2016)],[Taiwan Residents J.Math.,20(4),723–741(2016)],etc.,the current authors and T.C.Yang obtain explicit formulas for the numbers of superspecial abelian surfaces over finite fields.In this paper,we give an explicit formula for the size of the isogeny class of simple abelian surfaces with real Weil number q1/2.This establishes a key step that extends our previous explicit calculation of superspecial abelian surfaces to those of supersingular abelian surfaces.The second part is to introduce the notion of genera and idealcomplexes of abelian varieties with additional structures in a general setting.The purpose is to generalize the previous work by the second named author[Forum Math.,22(3),565–582(2010)]on abelian varieties with additional structures to similitude classes,which establishes more results on the connection between geometrically defined and arithmetically defined masses for further investigations.
文摘In this paper we prove that for any prime power q equivalent to 3 (mod 8) there exist 4 - {q(2); k, k, k, k; lambda} supplementary difference sets (SDSs) with k = q(q - 1)/2, lambda 4k - q(2), and Hadamard matrices of order 4q(2), and give several constructions of these SDSs. Moreover, combining the results of reference [1], we conclude that for any prime p equivalent to 3 (mod 8) and integer r greater than or equal to 1 there exists an Hadamard matrix of order 4p(2r).
基金supported by the MoST(Grant Nos.104-2115-M-001-001-MY3 and 107-2115-M-001-001-MY2)
文摘A theorem of Chow concerns homomorphisms of two abelian varieties under a primary field extension base change. In this paper, we generalize Chow’s theorem to semi-abelian varieties.This contributes to different proofs of a well-known result that every algebraic torus splits over a finite separable field extension. We also obtain the best bound for the degrees of splitting fields of tori.
文摘This paper discusses quantum mechanical schemas for describing waves with non-abelian phases, Fock spaces of annihilation-creation operators for these structures, and the Feynman recipe for obtaining descriptions of particle interactions with external fields.
文摘Quartic unramified Abelian extension fields of a class of cubic cyclic fields are given and the Hilbert class field of a cubic cyclic field with discriminant 607~2 iS obtained.
基金This work was supported by the National Fundamental Science Research Grant of China and the State Key Laboratory on Information Security.
文摘Several new results on non-existence of generalized bent functions are presented by using the class group of related imaginary abelian number fields.
