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
In this paper,we introduce a new concept,namelyε-arithmetics,for real vectors of any fixed dimension.The basic idea is to use vectors of rational values(called rational vectors)to approximate vectors of real values o...In this paper,we introduce a new concept,namelyε-arithmetics,for real vectors of any fixed dimension.The basic idea is to use vectors of rational values(called rational vectors)to approximate vectors of real values of the same dimension withinεrange.For rational vectors of a fixed dimension m,they can form a field that is an mth order extension Q(α)of the rational field Q whereαhas its minimal polynomial of degree m over Q.Then,the arithmetics,such as addition,subtraction,multiplication,and division,of real vectors can be defined by using that of their approximated rational vectors withinεrange.We also define complex conjugate of a real vector and then inner product and convolutions of two real vectors and two real vector sequences(signals)of finite length.With these newly defined concepts for real vectors,linear processing,such as linear filtering,ARMA modeling,and least squares fitting,can be implemented to real vectorvalued signals with real vector-valued coefficients,which will broaden the existing linear processing to scalar-valued signals.展开更多
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.
文摘In this paper,we introduce a new concept,namelyε-arithmetics,for real vectors of any fixed dimension.The basic idea is to use vectors of rational values(called rational vectors)to approximate vectors of real values of the same dimension withinεrange.For rational vectors of a fixed dimension m,they can form a field that is an mth order extension Q(α)of the rational field Q whereαhas its minimal polynomial of degree m over Q.Then,the arithmetics,such as addition,subtraction,multiplication,and division,of real vectors can be defined by using that of their approximated rational vectors withinεrange.We also define complex conjugate of a real vector and then inner product and convolutions of two real vectors and two real vector sequences(signals)of finite length.With these newly defined concepts for real vectors,linear processing,such as linear filtering,ARMA modeling,and least squares fitting,can be implemented to real vectorvalued signals with real vector-valued coefficients,which will broaden the existing linear processing to scalar-valued signals.
