scholarly journals HIGHER PRODUCT PYTHAGORAS NUMBERS OF SKEW FIELDS

2010 ◽  
Vol 03 (01) ◽  
pp. 193-207
Author(s):  
Dejan Velušček
Keyword(s):  
Laurent Series ◽  
Skew Field ◽  
Skew Fields ◽  
Pythagoras Number

We introduce the n–th product Pythagoras number p n(D), the skew field analogue of the n–th Pythagoras number of a field. For a valued skew field (D, v) where v has the property of preserving sums of permuted products of n–th powers when passing to the residue skew field k v and where Newton's lemma holds for polynomials of the form Xn - a, a ∈ 1 + I v , p n(D) is bounded above by either p n( k v ) or p n( k v ) + 1. Spherical completeness of a valued skew field (D, v) implies that the Newton's lemma holds for Xn - a, a ∈ 1 + I v but the lemma does not hold for arbitrary polynomials. Using the above results we deduce that p n (D((G))) = p n(D) for skew fields of generalized Laurent series.

Artin's problem for skew field extensions

1985 ◽  
Vol 97 (1) ◽  
pp. 1-6 ◽  
Author(s):  
A. H. Schofield
Keyword(s):  
Artinian Ring ◽  
Field Extension ◽  
Subsequent Paper ◽  
Commutative Field ◽  
Finite Representation ◽  
New Approach ◽  
Skew Field ◽  
Skew Fields ◽  
Left And Right ◽  
The Right

For a commutative field extension, L ⊃ K, it is clear that a left basis of L over K; is also a right basis of L over K; however, for an extension of skew fields, this may easily fail, though it is hard to determine whether the right and left dimension may be different. Cohn ([4], ch. 5), however, was able to find extensions of skew fields such that the left and right dimensions were an arbitrary pair of cardinals subject only to the restrictions that neither were 1 and at least one of them was infinite. In this paper, I shall present a new approach that allows us to construct extensions of skew fields such that the left and right dimensions are arbitrary integers not equal to 1. In a subsequent paper, [7], I shall present related results and consequences; in particular, there is a construction of a hereditary artinian ring of finite representation type corresponding to the Coxeter diagram I2(5) answering the question raised by Dowbor, Ringel and Simson[5].

scholarly journals Order in Affine and Projective Geometry

1963 ◽  
Vol 6 (1) ◽  
pp. 37-43 ◽  
Author(s):  
Joe Lipman
Keyword(s):  
Projective Geometry ◽  
Affine Geometry ◽  
Skew Field ◽  
Ordered Fields ◽  
Skew Fields

In this note we give a characterisation of ordered skew fields by certain properties of the negative elements.Properties of negative elements of a skew field K are then interpreted as statements about "betweenness" in any affine geometry over K, or as statements about "separation" in any projective geometry over K.In this way, our axioms for ordered fields permit of immediate translation into postulates of order for affine or projective geometry (over a field). The postulates so obtained seem simple enough to be of interest, (cf. [4], p. 22)

The word problem for free fields

1973 ◽  
Vol 38 (2) ◽  
pp. 309-314 ◽  
Author(s):  
P. M. Cohn
Keyword(s):  
Normal Form ◽  
Word Problem ◽  
Free Field ◽  
Free Associative Algebra ◽  
Commutative Field ◽  
Skew Field ◽  
Solvable Word Problem ◽  
Skew Fields ◽  
Free Fields ◽  
Solvable Word

It has long been known that every free associative algebra can be embedded in a skew field [11]; in fact there are many different embeddings, all obtainable by specialization from the ‘universal field of fractions’ of the free algebra (cf. [5, Chapter 7]). This makes it reasonable to call the latter the free field; see §2 for precise definitions. The existence of this free field was first established by Amitsur [1], but his proof is rather indirect and does not provide anything like a normal form for the elements of the field. Actually one cannot expect to find such a normal form, since it does not even exist in the field of fractions of a commutative integral domain, but at least one can raise the word problem for free fields: Does there exist an algorithm for deciding whether a given expression for an element of the free field represents zero?Now some recent work has revealed a more direct way of constructing free fields ([4], [5], [6]), and it is the object of this note to show how this method can be used to solve the word problem for free fields over infinite ground fields. In this connexion it is of interest to note that A. Macintyre [9] has shown that the word problem for skew fields is recursively unsolvable. Of course, every finitely generated commutative field has a solvable word problem (see e.g. [12]).The construction of universal fields of fractions in terms of full matrices is briefly recalled in §2, and it is shown quite generally for a ring R with a field of fractions inverting all full matrices, that if the set of full matrices over R is recursive, then the universal field has a solvable word problem. This holds more generally if the precise set of matrices over R inverted over the field is recursive, but it seems difficult to exploit this more general statement.

scholarly journals Skew fields with a non-trivial generalised power central rational identity

1994 ◽  
Vol 49 (1) ◽  
pp. 85-90 ◽  
Author(s):  
Katsuo Chiba
Keyword(s):  
Positive Integer ◽  
Skew Field ◽  
Finite Dimensional ◽  
Rational Identity ◽  
Skew Fields ◽  
Power Central

Let D be a skew field with uncountable centre K. The main result in the present paper is as follows: If D satisfies a non-trivial generalised power central rational identity, then D is finite dimensional over K. As a corollary we obtain the following result. Let a be an element of D such that (a−1x−1ax)q(x) ∈ K for all x ∈ D ﹨ {0} where q(x) is a positive integer depending on x. Then a ∈ K.

Prime Segments of Skew Fields

1995 ◽  
Vol 47 (6) ◽  
pp. 1148-1176 ◽  
Author(s):  
H. H. Brungs ◽  
M. Schröder
Keyword(s):  
Vector Spaces ◽  
Group Rings ◽  
Affine Transformations ◽  
Skew Field ◽  
Ordered Groups ◽  
Ordered Fields ◽  
Ordered Vector Spaces ◽  
Skew Fields ◽  
Dedekind Cuts ◽  
The Relationship

AbstractAn additive subgroup P of a skew field F is called a prime of F if P does not contain the identity, but if the product xy of two elements x and y in F is contained in P, then x or y is in P. A prime segment of F is given by two neighbouring primes P1 ⊃ P2; such a segment is invariant, simple, or exceptional depending on whether A(P1) = {a ∈ P1 | P1aP1 ⊂ P1} equals P1, P2 or lies properly between P1 and P2. The set T(F) of all primes of F together with the containment relation is a tree if |T(F)| is finite, and 1 < |T(F)| < ∞ is possible if F is not commutative. In this paper we construct skew fields with prescribed types of sequences of prime segments as skew fields F of fractions of group rings of certain right ordered groups. In particular, groups G of affine transformations on ordered vector spaces V are considered, and the relationship between properties of Dedekind cuts of V, certain right orders on G, and chains of prime segments of F is investigated. A general result in Section 4 describing the possible orders on vector spaces over ordered fields may be of independent interest.

scholarly journals Automorphisms of a Clifford-like parallelism

2021 ◽  
Vol 21 (1) ◽  
pp. 63-73
Author(s):  
Hans Havlicek ◽  
Stefano Pasotti ◽  
Silvia Pianta
Keyword(s):  
Automorphism Group ◽  
Skew Field ◽  
3 Dimensional ◽  
Skew Fields ◽  
Double Space

Abstract We focus on the description of the automorphism group Γ∥ of a Clifford-like parallelism ∥ on a 3-dimensional projective double space (ℙ(HF ), ∥ ℓ , ∥ r ) over a quaternion skew field H (of any characteristic). We compare Γ∥ with the automorphism group Γ ℓ of the left parallelism ∥ ℓ , which is strictly related to Aut(H). We build up and discuss several examples showing that over certain quaternion skew fields it is possible to choose ∥ in such a way that Γ∥ is either properly contained in Γ ℓ or coincides with Γ ℓ even though ∥ ≠ ∥ ℓ .

Skew Fields

1983 ◽  
Author(s):  
P. K. Draxl
Keyword(s):  
Skew Fields

Beta dynamical systems over the formal fields

2014 ◽  
Vol 51 (4) ◽  
pp. 454-465
Author(s):  
Lu-Ming Shen ◽  
Huiping Jing
Keyword(s):  
Dynamical Systems ◽  
Finite Field ◽  
Haar Measure ◽  
Laurent Series ◽  
Formal Laurent Series ◽  
Almost All

Let \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathbb{F}_q ((X^{ - 1} ))$$ \end{document} denote the formal field of all formal Laurent series x = Σ n=ν∞anX−n in an indeterminate X, with coefficients an lying in a given finite field \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathbb{F}_q$$ \end{document}. For any \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\beta \in \mathbb{F}_q ((X^{ - 1} ))$$ \end{document} with deg β > 1, it is known that for almost all \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$x \in \mathbb{F}_q ((X^{ - 1} ))$$ \end{document} (with respect to the Haar measure), x is β-normal. In this paper, we show the inverse direction, i.e., for any x, for almost all \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\beta \in \mathbb{F}_q ((X^{ - 1} ))$$ \end{document}, x is β-normal.

Sign in / Sign up

Export Citation Format

Share Document