ALGEBRAIC EXTENSION OF *-A OPERATOR

In this paper, we study various properties of algebraic extension of ＊-A operator.Specifically, we show that every algebraic extension of ＊-A operator has SVEP and is isoloid.And if T is an algebraic extension of ＊-A operator, then Weyl＇s theorem holds for f（T）, where f is an analytic functions on some neighborhood of σ（T） and not constant on each of the components of its domain.

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.

Algebraic Points of Any Degree l with (l ≥ 9) over Q on the Affine Equation Curve C3 (11): y11 = x3(x-1)3

In this work, we use the finiteness of the Mordell-weil group and the Riemann Roch spaces to give a geometric parametrization of the set of algebraic points of any given degree over the field of rational numbers Q on curve C3 (11): y11 = x3 (x-1)3. This result is a special case of quotients of Fermat curves Cr,s (p) : yp = xr(x-1)s, 1 ≤ r, s, r + s ≤ p-1 for p = 11 and r = s = 3. The results obtained extend the work of Gross and Rohrlich who determined the set of algebraic points on C1(11)(K) of degree at most 2 on Q.

On Two Classes of Extended 3-Lie Algebras

In this paper, based on the existing research results, we obtain the unary extension 3-Lie algebras by one-dimensional extension of the known Lie algebra L. For two known 3-Lie algebras H, M, the (μ, ρ, β)-extension of H through M is given, and the necessary and sufficient conditions for the (μ, ρ, β)-extension algebra of H through M being 3-Lie algebra are obtained, and the structural characteristics and properties of these two kinds of extended 3-Lie algebras are given.

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.

Dorroh extensions of algebras and coalgebras

We study Dorroh extensions of algebras and Dorroh extensions of coalgebras.Their structures are described.Some properties of these extensions are presented.We also introduce the finite duals of algebras and modules which are not necessarily unital.Using these finite duals,we determine the dual relations between the two kinds of extensions.

Frobenius bimodules and flat-dominant dimensions

We establish relations between Frobenius parts and between flat-dominant dimensions of algebras linked by Frobenius bimodules.This is motivated by the Nakayama conjecture and an approach of MartinezVilla to the Auslander-Reiten conjecture on stable equivalences.We show that the Frobenius parts of Frobenius extensions are again Frobenius extensions.Furthermore,let A and B be finite-dimensional algebras over a field k,and let domdim(_AX)stand for the dominant dimension of an A-module X.If_BM_A is a Frobenius bimodule,then domdim(A)domdim(_BM)and domdim(B)domdim(_AHom_B(M,B)).In particular,if B■A is a left-split(or right-split)Frobenius extension,then domdim(A)=domdim(B).These results are applied to calculate flat-dominant dimensions of a number of algebras:shew group algebras,stably equivalent algebras,trivial extensions and Markov extensions.We also prove that the universal(quantised)enveloping algebras of semisimple Lie algebras are QF-3 rings in the sense of Morita.

On the generalized derivations of bimodules

Recently, we introduced the notion of a generalized derivation from a bimodule to a bimodule. In this paper, we give a more general notion based on commutators which covers generalized derivations as a special case. Using it, we show that the separability of an algebra extension is characterized by generalized derivations.