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Given a collection A={L1,…,Ln} of intermediate fields in a field extension L/K of finite degree and ΛL,A=L1×⋯×Ln, there is a natural map ΨL,A:L→ΛL,A given by y↦(TrL/L1(y),…,TrL/Ln(y)), where TrL/Li...

This article investigates the cohomological kernels ker(H2n+1F→H2n+1E) of field extensions E/F in characteristic two where separable part of E is a quadratic extension F(α) and E is quadratic or qu...

Let G be a finite group, N and H subgroups of G with N

Let k be a field, K/k be a quadratic separable field extension, and 𝒜 a finite dimensional central simple algebra over K. If k is global or the field of fractions of a two-dimensional excellent henselian local domain with an algebraically closed residue field of characteristic zero and the degree of 𝒜 is odd, we prove that all K/k-involutions on 𝒜 are cyclic.

In characteristic zero, local monomialization is true along any valuation. However, we have recently shown that local monomialization is not always true in positive characteristic, even in two dimensional algebraic function fields. In this paper we show that local monomialization is true for defectless extensions of two dimensional excellent local rings, extending an earlier result of Piltant and the author for two dimensional algebraic function fields over an algebraically closed field. We also give theorems showing that in many cases there are good stable forms of the extension of associated graded rings in a finite separable field extension.

Let T be an integral domain with a maximal ideal M, ϕ: T → K: = T/M the natural surjection, and R the pullback ϕ−1(D), where D is a proper subring of K. We give necessary and sufficient conditions for the mixed extensions R[x 1]]…[x n ]] to be catenarian, where each [x i ]] is fixed as either [x i ] or [[x i ]]. We also give a complete answer to the question of determining the field extensions k ⊂ K such that the contraction map Spec(K[x 1]]…[x n ]]) → Spec(k[x 1]]…[x n ]]) is a homeomorphism. As an application, we characterize the globalized pseudo-valuation domains R such that R[x 1]]…[x n ]] is catenarian.

One of the most important results of Chevalley's extension theorem states that every valuation domain has at least one extension to every extension field of its quotient field. We state a generalization of this result for Prufer domains with any finite number of maximal ideals. Then we investigate extensions of semilocal Prufer domains in algebraic field extensions. In particular, we find an upper bound for the cardinality of extensions of a semilocal Prufer domain. Moreover, we show that any two extensions of a semilocal Prufer domain are incomparable (by inclusion) in an algebraic extension of fields.

An elementary proof using matrix theory is given for the following criterion: if F/K and L/K are field extensions, with F and L both contained in a common extension field, then F and L are linearly disjoint over K if (and only if) some K-vector space basis of F is linearly independent over L. The material in this note could serve as enrichment material for the unit on fields in a first course on abstract algebra.

We show in this article that in many cases the subfields of a nondegenerate tame semiramified division algebra of prime power degree over a Henselian valued field are inertial field extensions of t...

Let d be an odd integer, and let k be a field which contains a primitive dth root of unity. Let l 1 and l 2 be cyclic field extensions of k of degree d with norms n l 1/k and n l 2/k . Minac's approach which showed that quadratic Pfister forms are strongly multiplicative is applied to the form n l 1/k ⊗ n l 2/k of degree d. Let K = k(X 1,…, X d 2 ). We compute polynomials which are similarity factors of a form of the kind N ⊗ (n l 2/k ⊗ k K) over K, where N is the norm of a certain field extension of K of degree d. These polynomials arise by specializing certain indeterminates of the homogeneous polynomial representing the form n l 1/k ⊗ n l 2/k to be zero. Similar results are obtained for the tensor product of the norm of a cubic division algebra and a cubic norm n l 1/k .

We consider the ring Kr(F/D), where D is a subring of a field F, that is the intersection of the trivial extensions to F(X) of the valuation rings of the Zariski–Riemann space consisting of all valuation rings of the extension F/D and investigate the ideal structure of Kr(F/D) in the case where D is an affine algebra over a subfield K of F and the extension F/K has countably infinite transcendence degree, by using the topological structure of the Zariski–Riemann space. We show that for any pair of nonnegative integers d and h, there are infinitely many prime ideals of dimension d and height h that are minimal over any proper nonzero finitely generated ideal of Kr(F/D).

Let F be a field of characteristic 0 or greater than d. We investigate anisotropic forms of degree d > 2 and their behaviour under field extensions of degree coprime to d. A form of degree d over F which permits composition or Jordan composition remains anisotropic under any field extension of degree coprime to d. Other classes of forms are constructed which stay anisotropic under such extensions.

The commuting graph of a ring R, denoted by Γ(R), is a graph whose vertices are all noncentral elements of R, and two distinct vertices x and y are adjacent if and only if xy = yx. The commuting graph of a group G, denoted by Γ(G), is similarly defined. In this article we investigate some graph-theoretic properties of Γ(M n (F)), where F is a field and n ≥ 2. Also we study the commuting graphs of some classical groups such as GL n (F) and SL n (F). We show that Γ(M n (F)) is a connected graph if and only if every field extension of F of degree n contains a proper intermediate field. We prove that apart from finitely many fields, a similar result is true for Γ(GL n (F)) and Γ(SL n (F)). Also we show that for two fields F and E and integers n, m ≥ 2, if Γ(M n (F))≃Γ(M m (E)), then n = m and |F|=|E|.

For every field extension L/K, one has a restriction map res = res L/K : B(K) → B(L) of Brauer groups, sending a Brauer class [A] to [A⊗ K L]. (So actually, res is extension of scalars; it is calle...

We review the depth two and Hopf algebroid-Galois theory in math.RA/0108067 and specialize to induced representations of semisimple algebras and character theory of finite groups. We show that depth two subgroups over the complex numbers are normal subgroups. As a converse we observe that normal Hopf subalgebras over a field are depth two extensions. We introduce a generalized Miyashita-Ulbrich action on the centralizer of a ring extension, and apply it to a study of depth two and separable extensions, providing new characterizations of separable and H-separable extensions. With a view to the problem of when separable extensions are Frobenius, we supply a trace ideal condition for when a ring extension is Frobenius., final version, 19 pages. to appear: Communications in Algebra

Let k be a field, char k ≠ 2, F = k(x), D a biquaternion division algebra over k, and σ an orthogonal involution on D with nontrivial discriminant. We show that there exists a quadratic form ϕ ∈ I 2(F) such that dim ϕ = 8, [C(ϕ)] = [D], and ϕ does not decompose into a direct sum of two forms similar to two-fold Pfister forms. This implies in particular that the field extension F(D)/F is not excellent. Also we prove that if A is a central simple K-algebra of degree 8 with an orthogonal involution σ, then σ is hyperbolic if and only if σ K(A) is hyperbolic. Finally, let σ be a decomposable orthogonal involution on the algebra M 2 m (K). In the case m ≤ 5 we give another proof of the fact that σ is a Pfister involution. If m ≥ 2 n−2 − 2 and n ≥ 5, we show that q σ ∈ I n (K), where q σ is a quadratic form corresponding to σ. The last statement is founded on a deep result of Orlov et al. (2000) concerning generic splittings of quadratic forms.

We provide an axiomatic concept of Kronecker function rings and apply it to associate a Kronecker function ring to any integral domain D and any ideal system (star operation) on D. We investigate its behavior in algebraic field extensions and its connection with the defining valuation domains.

Suppose k is a field and is a separable Frobenius extension of k-algebras with trivial centralizer , Markov trace, and N a direct summand in M as N-bimodules. Let and be the successive endomorphism rings in a Jones tower (cf. Sec. 2). We define in Sec. 3 a depth 2 condition on this tower by requiring that a basis of freely generates as an M-module and a basis of freely generates as an -module. Then we provae in Sec. 4 that A and B have involutive strongly separable Hopf algebra structures dual to one another. As our main results, we prove in Sec. 5 that is a B-module algebra such that is the smash product ; in Sec. 6, that M is a A-module algebra such that is . We show that the actions involved are both outer. In Sec. 7, we prove that is a Hopf-Galois extension and point out a converse, thereby finding a non-commutative analogue of the classical theorem: a finite degree field extension is Galois if and only if it is separable and normal.

We study two invariants, ν(L/K) and λ(L/K), arising from a field extension L/K. By definition, ν(L/K) is the cardinality of the set of fields F such that K ⊆ F ⊆ L; and λ(L/K) is the supremum of the set of cardinal numbers arising as lengths of chains {Fi } of fields such that K ⊆ Fi ⊆ L. If L can be generated by one element over an infinite field K and 2 ≤ [L : K] = n < ∞, then ν(L/K) ≤ 2 n−2 + 1, with equality if L/K is Galois with the Klein four-group as Galois group. Despite positive results in the case of Abelian Galois groups, λ is not generally additive for finite-dimensional towers; moreover, maximal chains of intermediate fields can exhibit noncatenarian behavior if [L : K] = 12. The theory for infinite-dimensional algebraic extensions is strikingly different. For instance, for each infinite cardinal number N, there exists a field K of cardinality N and a chain of cardinality 2 N consisting of algebraic field extensions of K; in other words, if denotes an algebraic closure of K, then and, a forti...

Given an algebraically closed field k, a finitely generated field extension K of k and a proper subfield F of K with [K : F] < ∞, we discuss the question of whether or not F contains k.

This paper describes both the implementation and the mathematics of an Abstract Algebra project. The project begins with a seemingly simple question about a polynomial equation. It provides a concrete setting for exploring the relationships between polynomials, matrices, modular arithmetic, unique factorization, roots of unity, and order of group elements to help students understand the underlying abstract ring, field, and group theory.

We study the structure of the function field K of the quintic Fermat curve F(5) in the following way: Let K m be a g-maximal rational subfield of K. Then the field extension K/K m is obtained by the projection from F(5) to a line with a center p, ∈ F(5). By using this fact, we consider the field extension K/K m from several point of view.

A characterization of finite Hankei matrices is given and it is shown that such matrices arise naturally as matrix representations of scaled trace forms of field extensions and etale algebras. An algorithm is given for calculating the signature and the Hasse invariant of these scaled trace forms.

We prove that a formal power series in 1/x, whose coefficients are in a field extension of Q and are algebraically independent over Q, is differentially transcendental (i.e. not differentially algebraic) over this field extension. This is stated without proof in [2]. This result provides a source of functions analytic at ∞ that are not differentially algebraic over R. Such functions are of particular interest, because their germs belong to Hardy fields, but not to the class E of [1]-the intersection of all maximal Hardy fields.

Fix a ground field F. Let be a finite collection of finite degree separable field extensions of F. To study how the fields in К are related to each other, one should construct a joint splitting field F К and describe the Galois group Gal(F К /F) In this paper we describe the general appearance of Gal(F К /F) for arbi-trary F when the F i all have degree ≤ 6. Then we apply this description to three interesting related examples. First and second, we take ground fields the p-adic fields Q 2 and Q 3, and К the collection of all degree ≤ 6 fields. Third, we take ground field the rational number field Q and К the collection of all degree ≤ 6 fields with absolute discriminant of the form 2a3b This paper complements three of our previous papers. The paper [4] roughly speaking considers the Galois theory associated to a single field K of degree ≤ 6, whereas here we consider a finite collection of such K. The paper [5] lists all degree ≤ 6 fields over Q 2 and Q 3 . These tables form the basis of our local examples....

Introduction. Let L be a field. An additive mapping d : L −→ L is said to be a derivation of L if d(ab) = d(a)b+ ad(b) for any a, b ∈ L. If K ⊂ L is a field extension, then a derivation d of L is said to be a K-derivation if d(αa) = αd(a) for any α ∈ K and a ∈ L. If d is a derivation of L such that d(L) = L, then we say that d is epimorphic. (a) Let K ⊂ L be a finite dimensional field extension and let d be a Kderivation of L. Then L has a direct sum decomposition L = Ker d ⊕ L′, as a K-module, where L′ is K-isomorphic to d(L). Thus dimK L > dimK d(L) yields d(L) 6= L for any K-derivation d. (b) Assume now that L is a purely transcendental extension with infinite transcendental basis {xλ; λ ∈ Λ} over Q, the field of rational numbers. Since |L| = |Λ| (the cardinalities of L and Λ), we may put L = {aλ; λ ∈ Λ}. Then the Q-derivation d of L defined by d(xλ) = aλ is epimorphic. Thus it is natural to ask whether a field extension K ⊂ L has (or has not) an epimorphic K-derivation. The purpose of this paper is to study on this problem.

We define higher degree hyperbolic forms, analogous to the quadratic hyperbolic forms. We prove the following descent result. Let f be a form of degree d ≥ 3 over a field F of characteristic 0, and let K|f be a field extension. Then if f is equivalent over K to a hyperbolic form, f must already be equivalent to it over F. We also prove that in the monoid of equivalence classes of forms defined over F of a fixed degree d ≥ 3, under the tensor product, the submonoid generated by the equivalence classes of the hyperbolic forms is free. The proofs of these results involve the calculation of the centres and the Lie algebras of the higher degree hyperbolic forms. For the convenience of the reader we expound some of Harrison's seminal paper [5].

A new fringing field extension expression is presented for the resonant frequency of electrically thick rectangular microstrip antennas. The theoretical resonant frequency results obtained by using this new fringing field extension expression are in very good agreement with the experimental results available in the literature.

1. Parallel edges on polygons Certain geometrical problems may be formulated algebraically. The paradigm case is the solution of ruler and compass construction problems by means of the theory of field extensions. Yet other examples can be found. We will show here how two results on regular polygons are related to permutations of the elements of a finite abelian group. In this note, G denotes a finite abelian group written multiplicatively. In order to avoid the trivial cases we suppose that the order of G is at least two. A complete mapping of G is a permutation f of G such that

Definitions of normality and separability for unitary commutative ring extensions are given from the perspective of algebraic and transcendental ring extensions. Necessary and sufficient conditions are given for the set of subrings T i , R ⊆ T i ⊆ S, in a normal separable extension S: R in which the extensions S: T i are normal separable extensions, to map uniquely bijectively on the set of all subgroups of the Galois group of S: R.

Let Q and R be the fields of rational and real numbers respectively. Recall that a real number r is algebraic over the rationals if there is a polynomial p with coefficients in Q that has r as a root, i.e., that has p(r) = 0. Any college freshman can understand that idea, but things get more challenging when one asks about arithmetic with algebraic numbers. For example, being the roots of x2− 3 and x2− 20 respectively, the real numbers r1 = √ 3 and s1 = 2 √ 5 are certainly algebraic over the rationals, but what about the numbers r1 + s1, r1s1 and r1 s1 ? As it happens, all three are algebraic over the rationals. For example, r1 +s1 is a root of x4−46x2 +289. But how was that polynomial constructed, and what rationalcoefficient-polynomials have r1s1 and r1 s1 as roots? Students who take a second modern algebra course will learn to use field extension theory to show that the required polynomials must exist. They will learn that whenever r and s 6= 0 are algebraic over Q, then the field Q(r,s) is an extension of Q of finite degree with the consequence that r + s, rs and s are indeed algebraic over Q (see [2, 3, 7]). However, one would hope that students would encounter more elementary solutions for such basic arithmetic questions. Furthermore, one might want to know how to construct rational-coefficient polynomials that have r+s, rs and s as roots and thereby obtain bounds on the minimum degrees of such polynomials.

Let p be an odd prime. Then -1 is a quadratic residue (mod p) if and only if p 1 (mod 4), and 3 is a quadratic residue (mod p) if and only if p = 1 (mod 6). These results are well known. The first follows immediately from Euler's criterion and it appears as a theorem in just about every book in which quadratic residues are discussed. The proof of the result about -3 usually requires a little more effort and the law of quadratic reciprocity. The purpose of this note is to show that the "if' part of both results follows from Lagrange's theorem in group theory and the theory of algebraic field extensions. Let Zp be the field of integers (mod p). Assume that -1 is not a square (quadratic residue) in Zp. The quadratic extension Zp( -1) has p2 elements. Thus its group of nonzero elements Zp( -)* has p2_ 1 elements and the subgroup Zp* (the group of nonzero elements in Zp) has p 1 elements. The group we want to examine is the factor group Zp( -1)*/Zp*, with p + 1 elements. A direct calculation shows that the coset represented by 1 + has order 4. Thus 4 divides p + 1, which means that p -1 (mod 4). To prove the result about -3, extend Zp to Zp( -3). The order of the coset represented by 3 + -3 in the factor group Zp(-30)*/Zp* is 6. So, again by Lagrange's theorem, p -1 (mod 6).

If K/k is a finite, separable field extension K can be made to a quadratic space by adjoining the map for x∊K-the "trace form of K/k". In this paper I will determine all quadratic forms of an algebraic number field k which are isometric to a trace form of a normal (resp. abelian, resp. cyclic) field extension K/k.

In [5] it was shown that for a polynomial P of precise degree n with coefficients in an arbitrary m-ary algebra of dimension d as a vector space over an algebraically closed fields, the zeros of P together with the homogeneous zeros of the dominant part of P form a set of cardinality nd or the cardinality of the base field. We investigate polynomials with coefficients in a d dimensional algebra A without assuming the base field k to be algebraically closed. Separable polynomials are defined to be those which have exactly nd distinct zeros in [Ktilde] ⨷k A [Ktilde] where [Ktilde] denotes an algebraic closure of k. The main result states that given a separable polynomial of degree n, the field extension L of minimal degree over k for which L ⨷k A contains all nd zeros is finite Galois over k. It is shown that there is a non empty Zariski open subset in the affine space of all d-dimensional k algebras whose elements A have the following property: In the affine space of polynomials of precise degree n with co...