Algebraic Field Extension Research Articles

Roots of Polynomials in Algebraic Extensions of Fields

Roots of Polynomials in Algebraic Extensions of Fields

Roots of Polynomials in Algebraic Extensions of Fields

Roots of Polynomials in Algebraic Extensions of Fields

On multiplication in algebraic extension fields

In this paper we characterize all algorithms for obtaining the coefficients of ( Σ n−1 i=0 x i u i )( Σ n−1 i=0 y i u i ) mod P( u), where P( u) is an irreducible po lynomial of degree n, which use 2 n − 1 multiplications. It is shown that up to equivalence, all such algorithms are obtainable by first obtaining the coefficients of the product of two polynomials, and then reducing modulo the irreducible polynomial.

Automorphic forms and algebraic extensions of number fields, II

Automorphic forms and algebraic extensions of number fields, II

Copying configurations in algebraic extensions of prime fields

Copying configurations in algebraic extensions of prime fields

The prime ideals of a polynomial ideal under extension of the base field

Let k be a field, k′ an algebraic extension field, X 1 , ..., Xn indeterminates, A an ideal in k[X 1 , ..., Xn], and let A′ = k[X 1 , ..., Xn]A. It is shown that any associated prime of A′ contracts in k[X 1 , ..., Xn] to an associated prime of A. Some other old questions of Krull are also answered.

Automorphic forms and algebraic extensions of number fields

Automorphic forms and algebraic extensions of number fields

Exponentials in differentially algebraic extension fields

Exponentials in differentially algebraic extension fields

Algebraic Extensions of Difference Fields

An inversive difference field $\mathcal {K}$ is a field K together with a finite number of automorphisms of K. This paper studies inversive extensions of inversive difference fields whose underlying field extensions are separable algebraic. The principal tool in our investigations is a Galois theory, first developed by A. E. Babbitt, Jr. for finite dimensional extensions of ordinary difference fields and extended in this work to partial difference field extensions whose underlying field extensions are infinite dimensional Galois. It is shown that if $\mathcal {L}$ is a finitely generated separable algebraic inversive extension of an inversive partial difference field $\mathcal {K}$ and the automorphisms of $\mathcal {K}$ commute on the underlying field of $\mathcal {K}$ then every inversive subextension of $\mathcal {L}/\mathcal {K}$ is finitely generated. For ordinary difference fields the paper makes a study of the structure of benign extensions, the group of difference automorphisms of a difference field extension, and two types of extensions which play a significant role in the study of difference algebra: monadic extensions (difference field extensions $\mathcal {L}/\mathcal {K}$ having at most one difference isomorphism into any extension of $\mathcal {K}$) and incompatible extensions (extensions $\mathcal {L}/\mathcal {K},\mathcal {M}/\mathcal {K}$ having no difference field compositum).

Algebraic extensions of difference fields

An inversive difference field K \mathcal {K} is a field K together with a finite number of automorphisms of K. This paper studies inversive extensions of inversive difference fields whose underlying field extensions are separable algebraic. The principal tool in our investigations is a Galois theory, first developed by A. E. Babbitt, Jr. for finite dimensional extensions of ordinary difference fields and extended in this work to partial difference field extensions whose underlying field extensions are infinite dimensional Galois. It is shown that if L \mathcal {L} is a finitely generated separable algebraic inversive extension of an inversive partial difference field K \mathcal {K} and the automorphisms of K \mathcal {K} commute on the underlying field of K \mathcal {K} then every inversive subextension of L / K \mathcal {L}/\mathcal {K} is finitely generated. For ordinary difference fields the paper makes a study of the structure of benign extensions, the group of difference automorphisms of a difference field extension, and two types of extensions which play a significant role in the study of difference algebra: monadic extensions (difference field extensions L / K \mathcal {L}/\mathcal {K} having at most one difference isomorphism into any extension of K \mathcal {K} ) and incompatible extensions (extensions L / K , M / K \mathcal {L}/\mathcal {K},\mathcal {M}/\mathcal {K} having no difference field compositum).

Adams' operation and the symbol for the normed residue

We study the possibility of a meaningful extension of the ring of cohomologies of a point using algebraic extensions of local fields. It appears that complex K-theory is a good model problem for this. We prove that the effect of the Adams operation on the cohomologies of a point in extended K-theory coincides with the symbol for the local Artin reciprocity law. K-theories contained in a fixed K-theory with extended ring of cohomologies of a point are defined by higher branching groups.

Galois Theory of Essential Extensions of Modules

The purpose of this paper is to exploit an analogy between algebraic extensions of fields and essential extensions of modules, in which the role of the algebraic closure of a field F is played by the injective hull H(M) of a unitary left R-module M. (The notion of * ‘algebraic’ extensions of general algebraic systems has been studied by Shoda; see, for example [5].)In this analogy, the role of a polynomial p(x) is played by a homomorphism of R-modules(1)which will be called an ideal homomorphism into M. The process of solving the equation p(x) = 0 in F, or in an algebraic extension of F, will be replaced by the process of extending an ideal homomorphism (1) to a homomorphism F* from R into M, or into an essential extension of M.

Separability in an Algebra with Semi-Linear Homomorphism

The purpose of this paper is to outline a simple theory of separability for a non-associative algebra A with semi-linear homomorphism σ. Taking A to be a finite dimensional abelian Lie p-algebra L and σ to be the pth power operation in L, this separability is the separability of [2]. Taking A to be an algebraic field extension K over k and σ to be the Frobenius (pth power) homomorphism in K, this separability is the usual separability of K over k. The theory also applies to any unital non-associative algebra A over a field k and any unital homomorphism σ from A to A such that σ(ke) ⊂ ke, e being the identity element of A.

Integrally closed subrings of an integral domain

Let D be an integral domain with identity having quotient field K. This paper gives necessary and sufficient conditions on D in order that each integrally closed subring of D should belong to some subclass of the class of integrally closed domains; some of the subclasses considered are the completely integrally closed domains, Prufer domains, and Dedekind domains. 1. The class of integrally closed domains contains several classes of domains which are of fundamental importance in commutative algebra. Unique factorization domains, Krull domains, domains of finite character, Priifer domains, completely integrally closed domains, Dedekind domains, and principal ideal domains are examples of such subclasses of the class of integrally closed domains. This paper considers the problems of determining, conversely, necessary and sufficient conditions on an integral domain with identity in order that each of its integrally closed subrings should belong to some subclass of the class of integrally closed domains. An example of a typical result might be Theorem 2.3: If J is an integral domain with identity having quotient field K, then conditions (1) and (2) are equivalent. (1) Each integrally closed subring of J is completely integrally closed. (2) Either J has characteristic 0 and K is algebraic over the field of rational numbers or J has characteristic p # 0 and K has transcendence degree at most one over its prime subfield. If J is integrally closed, then conditions (1) and (2) are equivalent to: (3) Each integrally closed subring of J with quotient field K is completely integrally closed. In considering characterizations of integral domains with identity for which every integrally closed subring is Dedekind or almost Dedekind (?3), we are led to use some results of W. Krull to prove Theorem 4.1, which establishes the existence of, as well as a method for constructing, a field with certain specified valuations. We then use this theorem to construct an example of an infinite separable algebraic extension field K of FLp(X) such that the integral closure J of FLp[X] in K Received by the editors April 7, 1970. AMS 1969 subject classifications. Primary 1315, 1350; Secondary 1320.

On Primitive Elements in Differentially Algebraic Extension Fields

On Primitive Elements in Differentially Algebraic Extension Fields

Existence of Non-Trivial Deformations of Inseparable Algebraic Extension Fields II

Let K be an extension of a field k, and p denotes the characteristic. It was proved by M. Gerstenhaber ([1]) that if K is separable over k, then it is rigid and it was conjectured in [1] that, if K is not separable over k, then it is not rigid. We studied in [4] the above conjecture in certain special case. In this note we shall extend the results of [4] to inseparable algebraic extension fields.

Algebraic Extensions of Fields.

Algebraic Extensions of Fields.

Note on G. Whaples’ paper “Algebraic extensions of arbitrary fields”

Note on G. Whaples’ paper “Algebraic extensions of arbitrary fields”

On primitive elements in differentially algebraic extension fields

It is well known that if F is a field of characteristic zero and K= F(cq, ..., ,Cc) is a finite algebraic extension of F, then K contains a primitive element, i.e. an element a such that F(al, . . ., a() = F(x). Moreover, by means of Galois theory, it is possible to characterize those elements of the extension field which are primitive. In the case of finite differentially algebraic extensions the theorem without further restrictions is false. Let Q be the field of rational numbers and 8 the usual derivation, i.e., 8q=O for every q E Q. Let c1, . . ., cn, be algebraically independent complex numbers over Q. If (Q , 8) is the differentially algebraic extension of Q where 8c=0 for every c E Q , then the underlying set of Q is identical with that of Q(c1, . . ., cs), whence it is clear that there is no element c E Q such that Q = Q . Kolchin [2] (also [5, p. 52]) has shown the existence of primitive elements in the case where the differential field F has one derivation operator and the field F has an element f such that 8f# 0(1). The differential fields (F , D) considered in this paper are differentially algebraic over F, but F does not contain nonconstant elements. We prove the existence of primitive elements in the case where the derivation operator satisfies the conditions

On automorphisms of polyadic algebras

Introduction. This paper belongs to the theory of polyadic algebras as developed by Halmos [11-15], but it has a bearing on the theory of models. Two central concepts are those of homogeneous(2) and of normal extensions of a polyadic algebra (see the beginning of ?2 and of ?6). These concepts bear some resemblance to concepts of the theory of algebraic extensions of fields; however, we have been influenced more immediately by M. Krasner's general Galois theory [33-35] which can be given a polyadic interpretation. Aside from the short preliminary Chapter 0, the paper is divided into sections grouped into three chapters. Each section begins by an outline of its contents. We proceed to an analysis of the main results of the paper. All polyadic algebras considered are locally finite of infinite degree. The highlights of Chapter I are: (i) the possibility of extending any simple extension of a (simple polyadic) algebra to a simple homogeneous and normal extension (Corollary 4.6); (ii) a first step on a Galois theory (Theorem 4.5) which, in the case of a full simple functional algebra with finite domain, takes a definite form (Theorem 4.7) closely related to Krasner's theory in that case; (iii) the existence and unicity of o+-universal-homogeneous algebras in the sense of B. Jonsson [21; 22] in certain classes of polyadic algebras (Theorem 6.3, Theorem 6.6). The results in (iii) make use of Jonsson's work which is dependent on the continuum hypothesis (see also Theorem 5.5 which is independent of this hypothesis). Chapter If departs from the theme of automorphisms and uses little aside from Chapter 0 and ?1. The main result here is Theorem 8.1 which gives a description of the functional representations of a full simple functional algebra in terms of a new generalization of the concept of reduced power. In Chapter III, we propose to show that the algebraic results of the first two chapters, when put together, embody several known model-theoretic results or new forms of such results. After a section devoted to the connections between Model theory and polyadic algebras, we give new proofs of Beth's theorem