An Introduction to the Theory of Field Extensions

This paper unfolds and reviews the theory of abstract algebra, field extensions and discusses various kinds of field extensions. Field extensions are said to be algebraic or transcendental. We pay much attention to algebraic extensions. Finally, we construct finite extensions of Q and finite extensions of the function field over finite field Fp using the notion of field completion, analogous to field extensions. With the study of field extensions, considering any polynomial with coefficients in the field, we can find the roots of the polynomial, and with the notion of algebraically closed fields, we have one field, F, where we can find the roots of any polynomial with coefficients in F.

Number of primitive elements of field extension GF(p^(nm))/GF(p^n)

Presents the counting of the counts number of primitive elements of finite dimensional field extension GF(p nm )/GF(p n) using (1) the principle of inclusion exclusion, (2) the Mbius inversion, (3) the Euler  function, and the new identity obtained ∑t｜p nm-1 , t p nmq j -1(t)=p nm -∑jp nmq j +∑j 1<j 2p nmqj 1qj 2 -…+(-1) kp nmq 1…q k  where, m>1, p is a prime, (·) is Euler function, and q 1,…,q k are the all distinct prime divisors of m.

An Infinite Family of Number Fields with No Inert Primes

The goal of this paper is to show that there are infinitely many number fields K/Q, for which there is no inert prime p ∈ N*, i.e. ∀p ∈ N* a prime number, prime ideal of K such that where: Zk is the Dedekind domain of the integer elements of K. To prove such a result, consider for any prime p, the decomposition into a product of prime ideals of Zk, of the ideal . From this point, we use on the one hand: 1) The well- known property that says: If , then the ideal pZk decomposes into a product of prime ideals of Zk as following: . (where:;is the irreducible polynomial of θ, and, is its reduction modulo p, which leads to a product of irreducible polynomials in Fp[X]). It is clear that because if is reducible in Fp[X], then consequently p is not inert. Now, we prove the existence of such p, by proving explicit such p as follows. So we use on the other hand: 2) this property that we prove, and which is: If , is an irreducible normalized integer polynomial, whose splitting field is , then for any prime number p ∈ N: is always a reducible polynomial. 3) Consequently, and this closes our proof: let’s consider the set (whose cardinality is infinite) of monogenic biquadratic number fields: . Then each fθ(X) checks the above properties, this means that for family M, all its fields, do not admit any inert prime numbers p ∈ N. 2020-Mathematics Subject Classification (MSC2020) 11A41 - 11A51 - 11D25 - 11R04 - 11R09 - 11R11 - 11R16 - 11R32 - 11T06 - 12E05 - 12F05 -12F10 -13A05-13A15 - 13B02 - 13B05 - 13B10 - 13B25 -13F05

Addition Sequence Method of Scalar Multiplication of Elliptic Curve over OEF

A new elliptic curve scalar multiplication algorithm is proposed. Thealgorithm uses the Frobenius map on optimal extension field (OEF) and addition sequence We introducea new algorithm on generating addition sequence efficiently and also give some analysis about it.Based on this algorithm, a new method of computing scalar multiplication of elliptic curve over anOEF is presented. The new method is more efficient than the traditional scalar multiplicationalgorithms of elliptic curve over OEF. Thecomparisons of traditional method and the new method arealso given.

A Heuristic Method of Scalar Multiplication of Elliptic Curve over OEF

Elliptic curve cryptosystem is the focus of public key cryptology nowadays, for it has many advantages RSA lacks. This paper introduced a new heuristic algorithm on computing multiple scalar multiplications of a given point. Based on this algorithm, a new method of computing scalar multiplication of elliptic curve over optimal extension field (OEF) using Frobenius map was presented. The new method is more efficient than the traditional ones. In the last part of this paper, the comparison was given in the end.

Multilayer Multidi mensional Extension Set Theory

In brder to study the contradiction problem of multilayer multidimensional complex systems, the concepts of extension field and stable field of intersection and union of multilayer multidimensional extension set are given. Then the related operations and properties are discussed. The results of study expand the concepts of intersection and union of extension set to a general situation, and provide the theoretical basis for production of the concepts of intersection and union of multilayer multidimensional matter element system extension set. In this way, it will be possible that matter element system theory is used to creative designs of complex systems.

ON SOME KEY GEOLOGICAL PROBLEMS RELATED TO THE LINGLONG-JIAOJIA ORE-CENTRALIZED DISTRICT IN SHANDONG PROVINCE,CHINA

The Linglong-Jiaojia ore-centralized district is controlled by the tectonic stress field characterized by the combination of extension and strike-slip, and the dip, dip angle, pitch and pitch angle of the ore bodies are all constrained by the dynamic conditions of the tectonics. The metallotectonic series for the ore-centralized district belong to the type of a combination of extension and strike-slip and can be subdivided into four sub-series. The ore-forming process in the brittle regime can be disintegrated into two stages, i.e., the embryonic fracture stage and the megascopic fracture stage, and ore-forming process is rather common in the ore-centralized district at the former stage. Moreover, several key structural patterns and their features were discussed and a preliminary assessment about the ore-forming prospect in this district was made in the paper.

Generation and Application of Spatial Atmospheric Turbulence Field in Flight Simulation

This article deals with generation and application of three-dimensional (3D) atmospheric turbulence field in large aircraft real-time flight simulation. The modeling requirements for the turbulence field of large aircraft flight simulation are analyzed here. The spatial turbulence field is generated in the frequency domain by using the Monte Carlo method,and then transformed back to the time domain with the 3D inverse Fourier transform. The von Karman model is adopted for an accurate description of the turb...

An efficient algorithm for factoring polynomials over algebraic extension field

An efficient algorithm is proposed for factoring polynomials over an algebraic extension field defined by a polynomial ring modulo a maximal ideal. If the maximal ideal is given by its CrSbner basis, no extra Grbbner basis computation is needed for factoring a polynomial over this extension field. Nothing more than linear algebraic technique is used to get a characteristic polynomial of a generic linear map. Then this polynomial is factorized over the ground field. From its factors, the factorization of the polynomial over the extension field is obtained. The algorithm has been implemented in Magma and computer experiments indicate that it is very efficient, particularly for complicated examples.

ε-arithmetics for real vectors and linear processing of real vector-valued signals

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.

Synthesis of Two-Color Laser Pulses for the Harmonic Cutoff Extension

Increasing simultaneously both the cutoff energy and efficiency is a big challenge to all applications of high-order harmonic generation(HHG).For this purpose,the shaping of the waveform of driving pulse is an alternative approach.Here,we show that the harmonic cutoff can be extended by about two times without reducing harmonic yield after considering macroscopic propagation effects,by adopting a practical way to synthesize two-color fields with fixed energy.Our results,combined with the experimental techniques,show the great potential of HHG as a tabletop light source.

Optimal Algorithm for Algebraic Factoring

This paper presents an optimized method for factoring multivariate polynomials over algebraic extension fields defined by an irreducible ascending set.The basic idea is to convert multivariate polynomials to univariate polynomials and algebraic extension fields to algebraic number fields by suitable integer substitutions. Then factorize the univariate polynomials over the algebraic number fields. Finally, construct multivariate factors of the original polynomial by Hensel lemma and TRUEFACTOR test. Some examples with timing are included.