scholarly journals Ring and Field Adjunctions, Algebraic Elements and Minimal Polynomials

2020 ◽  
Vol 28 (3) ◽  
pp. 251-261
Author(s):  
Christoph Schwarzweller
Keyword(s):  
Vector Space ◽  
Minimal Polynomial ◽  
Field Extension ◽  
Splitting Field ◽  
Minimal Polynomials ◽  
Algebraic Element ◽  
Algebraic Elements

Summary In [6], [7] we presented a formalization of Kronecker’s construction of a field extension of a field F in which a given polynomial p ∈ F [X]\F has a root [4], [5], [3]. As a consequence for every field F and every polynomial there exists a field extension E of F in which p splits into linear factors. It is well-known that one gets the smallest such field extension – the splitting field of p – by adjoining the roots of p to F. In this article we start the Mizar formalization [1], [2] towards splitting fields: we define ring and field adjunctions, algebraic elements and minimal polynomials and prove a number of facts necessary to develop the theory of splitting fields, in particular that for an algebraic element a over F a basis of the vector space F (a) over F is given by a 0 , . . ., an− 1, where n is the degree of the minimal polynomial of a over F .

Asymptotic representation of weighted L∞- and L1-minimal polynomials

2008 ◽  
Vol 144 (1) ◽  
pp. 241-254 ◽  
Author(s):  
ANDRÁS KROÓ ◽  
FRANZ PEHERSTORFER
Keyword(s):  
Weight Function ◽  
Chebyshev Polynomial ◽  
Asymptotic Representation ◽  
Minimal Polynomial ◽  
Maximum Norm ◽  
Weight Functions ◽  
Second Derivative ◽  
Minimal Polynomials ◽  
Minimum Deviation ◽  
Point Of Interest

AbstractIn 1858 Chebyshev, and some years later his students Korkin and Zolotarev, determined the polynomial which deviates least from zero among all polynomials of degree n with leading coefficient one with respect to the maximum- and the L1-norm, respectively; these are now called the Chebyshev polynomial of first and second kind.The next natural step which is to find, at least asymptotically, the minimal polynomial with respect to a given weight function has not been settled until today. Indeed, Bernstein gave asymptotics for the minimum deviation of weighted minimal polynomials, Fekete and Walsh found nth root asymptotics and, recently, Lubinsky and Saff provided asymptotics outside [−1, 1]. But the main point of interest: the asymptotic representation of the weighted minimal polynomials on the interval of approximation [−1, 1] remained open. Here we settle this problem with respect to the maximum norm for weight functions whose second derivative is Lipα, α ∈ (0, 1), and with respect to the L1-norm under somewhat stronger differentiability conditions.

scholarly journals Algebraic Extensions

2021 ◽  
Vol 29 (1) ◽  
pp. 39-47
Author(s):  
Christoph Schwarzweller ◽  
Agnieszka Rowińska-Schwarzweller
Keyword(s):  
Field Theory ◽  
Field Extension ◽  
Field Extensions ◽  
Intermediate Field ◽  
Finite Set ◽  
Algebraic Elements ◽  
Roots Of A Polynomial

Summary In this article we further develop field theory in Mizar [1], [2], [3] towards splitting fields. We deal with algebraic extensions [4], [5]: a field extension E of a field F is algebraic, if every element of E is algebraic over F. We prove amongst others that finite extensions are algebraic and that field extensions generated by a finite set of algebraic elements are finite. From this immediately follows that field extensions generated by roots of a polynomial over F are both finite and algebraic. We also define the field of algebraic elements of E over F and show that this field is an intermediate field of E|F.

On Asano’s theorem

2019 ◽  
Vol 18 (10) ◽  
pp. 1950181
Author(s):  
Münevver Pınar Eroǧlu ◽  
Tsiu-Kwen Lee ◽  
Jheng-Huei Lin
Keyword(s):  
Vector Space ◽  
Division Algebra ◽  
Division Algebras ◽  
Infinite Field ◽  
Additive Subgroup ◽  
Normal Bases ◽  
Algebraic Elements ◽  
Inner Automorphisms

Let [Formula: see text] be a division algebra over an infinite field [Formula: see text] such that every element of [Formula: see text] is a sum of finitely many algebraic elements. As a generalization of Asano’s theorem, it is proved that every noncentral subspace of [Formula: see text] invariant under all inner automorphisms induced by algebraic elements contains [Formula: see text], the additive subgroup of [Formula: see text] generated by all additive commutators of [Formula: see text]. From the viewpoint we study the existence of normal bases of certain subspaces of division algebras. It is proved among other things that [Formula: see text] is generated by multiplicative commutators as a vector space over the center of [Formula: see text].

scholarly journals SIMETRISASI BENTUK KANONIK JORDAN

2021 ◽  
Vol 15 (1) ◽  
pp. 015-028
Author(s):  
Darlena Darlena ◽  
Ari Suparwanto
Keyword(s):  
Scalar Field ◽  
Linear Operator ◽  
Characteristic Polynomial ◽  
Canonical Form ◽  
Minimal Polynomial ◽  
Splitting Field ◽  
Jordan Decomposition ◽  
Jordan Canonical Form ◽  
Canonical Matrix

If the characteristic polynomial of a linear operator  is completely factored in scalar field of  then Jordan canonical form  of  can be converted to its rational canonical form  of , and vice versa. If the characteristic polynomial of linear operator  is not completely factored in the scalar field of  ,then the rational canonical form  of  can still be obtained but not its Jordan canonical form matrix . In this case, the rational canonical form  of  can be converted to its Jordan canonical form by extending the scalar field of  to Splitting Field of minimal polynomial   of , thus forming the Jordan canonical form of  over Splitting Field of  . Conversely, converting the Jordan canonical form  of  over Splitting Field of  to its rational canonical form uses symmetrization on the Jordan decomposition basis of  so as to form a cyclic decomposition basis of  which is then used to form the rational canonical matrix of

Soft Int-Field Extension

2021 ◽  
pp. 1-25
Author(s):  
Jayanta Ghosh ◽  
Dhananjoy Mandal ◽  
Tapas Kumar Samanta
Keyword(s):  
Field Extension ◽  
Field Extensions ◽  
Algebraic Element

The relation between soft element-wise field and soft int-field has been established and then some properties of soft int-field are studied. We define the notions of soft algebraic element and soft purely inseparable element of a soft int-field extension. Some characterizations of soft algebraic and soft purely inseparable int-field extensions are given. Lastly, we define soft separable algebraic int-field extension and study some of its properties.

Topologies Extending Valuations

1978 ◽  
Vol 30 (01) ◽  
pp. 164-169 ◽  
Author(s):  
Thomas Rigo ◽  
Seth Warner
Keyword(s):  
Vector Space ◽  
Cartesian Product ◽  
Field Extension ◽  
Valuation Theory ◽  
Topological Isomorphism ◽  
Product Topology ◽  
Hausdorff Topology ◽  
Absolute Value ◽  
Finite Dimensional ◽  
Multilinear Mapping

Let K be a field complete for a proper valuation (absolute value) v. It is classic that a finite-dimensional K-vector space E admits a unique Hausdorff topology making it a topological K-vector space, and that that topology is the “cartesian product topology” in the sense that for any basis c1 …, cn of E, is a topological isomorphism from K n to E [1, Chap. I, § 2, no. 3; 2, Chap. VI, § 5, no. 2]. It follows readily that any multilinear mapping from E m to a Hausdorff topological K-vector space is continuous. In particular, any multiplication on E making it a K-algebra is continuous in both variables. If for some such multiplication E is a field extension of K, then by valuation theory the unique Hausdorff topology of E is given by a valuation (absolute value) extending v.

Strongly and co-strongly minimal abelian structures

2010 ◽  
Vol 75 (2) ◽  
pp. 442-458 ◽  
Author(s):  
Ehud Hrushovski ◽  
James Loveys
Keyword(s):  
Vector Space ◽  
Finite Index ◽  
Division Ring ◽  
First Case ◽  
Special Cases ◽  
Structure Theorems ◽  
Algebraic Element ◽  
Minimal Module ◽  
Induced Structure

AbstractWe give several characterizations of weakly minimal abelian structures. In two special cases, dual in a sense to be made explicit below, we give precise structure theorems:1. when the only finite 0-definable subgroup is {0}, or equivalently 0 is the only algebraic element (the co-strongly minimal case);2. when the theory of the structure is strongly minimal.In the first case, we identify the abelian structure as a “near-subspace” A of a vector space V over a division ring D with its induced structure, with possibly some collection of distinguished subgroups of A of finite index in A and (up to acl(∅)) no further structure. In the second, the structure is that of V/A for a vector space and near-subspace as above, with the only further possible structure some collection of distinguished points. Here a near-subspace of V is a subgroup A such that for any nonzero d ∈ D. the index of A ∩ dA, in A is finite. We also show that any weakly minimal abelian structure is a reduct of a weakly minimal module.

On a Theorem of Hermite and Joubert

1999 ◽  
Vol 51 (1) ◽  
pp. 69-95 ◽  
Author(s):  
Zinovy Reichstein
Keyword(s):  
Division Algebra ◽  
Minimal Polynomial ◽  
Field Extension ◽  
Division Algebras ◽  
Classical Theorem

AbstractA classical theorem of Hermite and Joubert asserts that any field extension of degree n = 5 or 6 is generated by an element whose minimal polynomial is of the form λn + c1λn−1 + ··· + cn−1λ + cn with c1 = c3 = 0. We show that this theorem fails for n = 3m or 3m + 3l (and more generally, for n = pm or pm + pl, if 3 is replaced by another prime p), where m > l ≥ 0. We also prove a similar result for division algebras and use it to study the structure of the universal division algebra UD(n).We also prove a similar result for division algebras and use it to study the structure of the universal division algebra UD(n).

Norm and spectral radius for algebraic elements of a Banach algebra

1980 ◽  
Vol 88 (1) ◽  
pp. 129-133 ◽  
Author(s):  
N. J. Young
Keyword(s):  
Banach Algebra ◽  
Spectral Radius ◽  
Upper Bound ◽  
Good Deal ◽  
Additional Information ◽  
Algebraic Element ◽  
Algebraic Elements ◽  
Image Position ◽  
Spectral Radius Formula ◽  
Unit Norm

The purpose of this note is to show that, for any algebraic element a of a Banach algebra and certain analytic functions f, one can give an upper bound for ‖f(a)‖ in terms of ‖a‖ and the spectral radius ρ(a) of a. To illustrate the nature of the result, consider the norms of powers of an element a of unit norm. In general, the spectral radius formulacontains all that can be said (that is, the limit ρ(a) can be approached arbitrarily slowly). If we have the additional information that a is algebraic of degree n we can say a good deal more. In the case of a C*-algebra we have the neat result that, if ‖a‖ ≤ 1,(see Theorem 2), while for a general Banach algebra we have at least

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