Basis Theorem

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Jonathan Brundan - One of the best experts on this subject based on the ideXlab platform.

Sophie Huczynska - One of the best experts on this subject based on the ideXlab platform.

  • The strong primitive normal Basis Theorem
    2010
    Co-Authors: Stephen D. Cohen, Sophie Huczynska
    Abstract:

    An element α of the extension E of degree n over the finite field F = GF(q) is called free over F if {α, α q , . . . , α q n 1 } is a (normal) Basis of E/F. The primitive normal Basis Theorem, first established in full by Lenstra and Schoof (1987), asserts that for any such extension E/F, there exists an element α ∈ E such that α is simultaneously primitive (i.e., generates the multiplicative group of E) and free over F. In this paper we prove the following strengthening of this Theorem: aside from four specific extensions E/F, there exists an element α ∈ E such that both α and α −1 are simultaneously primitive and free over F. AMS classification: Primary 11T30; Secondary 11T06, 12E20

  • The Strong Primitive Normal Basis Theorem
    2006
    Co-Authors: Stephen D. Cohen, Sophie Huczynska
    Abstract:

    An element w of the extension E of degree n over the finite field F=GF(q) is called free over F if {w, w^q,...,w^{q^{n-1}}} is a (normal) Basis of E/F. The Primitive Normal Basis Theorem, first established in full by Lenstra and Schoof (1987), asserts that for any such extension E/F, there exists an element w in E such that w is simultaneously primitive (i.e., generates the multiplicative group of E) and free over F. In this paper we prove the following strengthening of this Theorem: aside from five specific extensions E/F, there exists an element w in E such that both w and w^{-1} are simultaneously primitive and free over F.

  • the primitive normal Basis Theorem without a computer
    2003
    Co-Authors: Stephen D. Cohen, Sophie Huczynska
    Abstract:

    Given $q$ , a power of a prime $p$ , denote by $F$ the finite field ${\rm GF}(q)$ of order $q$ , and, for a given positive integer $n$ , by $E$ its extension ${\rm GF}(q^n)$ of degree $n$ . A primitive element of $E$ is a generator of the cyclic group $E^\ast$ . Additively too, the extension $E$ is cyclic when viewed as an $FG$ -module, $G$ being the Galois group of $E$ over $F$ . The classical form of this result – the normal Basis Theorem – is that there exists an element $\alpha \in E$ (an additive generator) whose conjugates $\{\alpha, \alpha^q, \ldots, \alpha^{q^{n-1}}\}$ form a Basis of $E$ over $F; \alpha$ is a free element of $E$ over $F$ , and a Basis like this is a normal Basis over $F$ . The core result linking additive and multiplicative structure is that there exists $\alpha \in E$ , simultaneously primitive and free over $F$ . This yields a primitive normal Basis over $F$ , all of whose members are primitive and free. Existence of such a Basis for every extension was demonstrated by Lenstra and Schoof [ 5 ] (completing work by Carlitz [ 1 , 2 ] and Davenport [ 4 ]).

  • the primitive normal Basis Theorem without a computer
    2003
    Co-Authors: Stephen D. Cohen, Sophie Huczynska
    Abstract:

    Given , a power of a prime , denote by the finite field of order , and, for a given positive integer , by its extension of degree . A primitive element of is a generator of the cyclic group . Additively too, the extension is cyclic when viewed as an -module, being the Galois group of over . The classical form of this result – the normal Basis Theorem – is that there exists an element (an additive generator) whose conjugates form a Basis of over is a free element of over , and a Basis like this is a normal Basis over . The core result linking additive and multiplicative structure is that there exists , simultaneously primitive and free over . This yields a primitive normal Basis over , all of whose members are primitive and free. Existence of such a Basis for every extension was demonstrated by Lenstra and Schoof [5] (completing work by Carlitz [1, 2] and Davenport [4]).

Jonathan Comes - One of the best experts on this subject based on the ideXlab platform.

Andrew Reynolds - One of the best experts on this subject based on the ideXlab platform.

Ross Willard - One of the best experts on this subject based on the ideXlab platform.

  • Congruence meet-semidistributive locally finite varieties and a finite Basis Theorem
    2018
    Co-Authors: George F. Mcnulty, Ross Willard
    Abstract:

    We provide several conditions that, among locally finite varieties, characterize congruence meet-semidistributivity and we use these conditions to give a new proof of a finite Basis Theorem published by Baker, McNulty, and Wang in 2004. This finite Basis Theorem extends Willard’s Finite Basis Theorem.

  • A finite Basis Theorem for difference-term varieties with a finite residual bound
    2015
    Co-Authors: Keith A. Kearnes, Ágnes Szendrei, Ross Willard
    Abstract:

    We prove that if V is a variety of algebras (i.e., an equationally axiomatizable class of algebraic structures) in a finite language, V has a difference term, and V has a finite residual bound, then V is finitely axiomatizable. This provides a common generalization of R. McKenzie’s finite Basis Theorem for congruence modular varieties with a finite residual bound, and R. Willard’s finite Basis Theorem for congruence meet-semidistributive varieties with a finite residual bound. This paper is a contribution to an old problem in logic from the schools of A. Tarski and A. Maltsev: which finite algebraic structures A (algebras for short) have a finite Basis for their identities? Equivalently, for which finite algebras A is the variety V(A) (the smallest equational class containing A) finitely axiomatizable? On the one hand, every finite group [38], finite ring [22, 28], finite commutative semigroup [40], finite lattice [33], or two-element algebra in a finite language [30] is known to be finitely based. On the other hand, the list of finite algebras which are not finitely based includes, in addition to pathological examples (e.g., [31, 37]), some finite semigroups [40], some finite non-associative K-algebras [41, 29, 12], and even a finite group with one non-identity element named by a constant [5]. In 1996 R. McKenzie [36] proved that the problem of determining whether a finite algebra is finitely based is undecidable, settling Tarski’s finite Basis problem. The evidence suggests that a full classification of finitely based finite algebras is beyond reach. However, there are some remarkable partial results. In particular, in the early 1970s K. Baker [1, 2] proved the following: if A is a finite algebra in a finite language and V(A) is congruence distributive (i.e., for every B ∈ V(A), the lattice of congruence relations of B is a distributive lattice), then A is finitely based. Two important ingredients in the proof were provided by B. Jonsson [13]: (1) a characterization, in terms of identities, of the condition that a variety be congruence distributive, and (2) a proof that if A is finite and V(A) is congruence distributive, then every subdirectly Date: July 8, 2014. 2010 Mathematics Subject Classification. Primary 03C05; Secondary 08B05.

  • a finite Basis Theorem for residually finite congruence meet semidistributive varieties
    2000
    Co-Authors: Ross Willard
    Abstract:

    We derive a Mal'cev condition for congruence meet-semidistributivity and then use it to prove two Theorems. Theorem A: if a variety in a finite language is congruence meet-semidistributive and residually less than some finite cardinal, then it is finitely based. Theorem B: there is an algorithm which, given m < ω and a finite algebra in a finite language, determines whether the variety generated by the algebra is congruence meet-semidistributive and residually less than m.

  • a finite Basis Theorem for residually finite congruence meet semidistributive varieties
    1997
    Co-Authors: Ross Willard
    Abstract:

    We derive a Mal'cev condition for congruence meet-semidistributivity and then use it to prove two Theorems. Theorem A: if a variety in a finite language is congruence meet-semidistributive and residually less than some finite cardinal, then it is finitely based. Theorem B: there is an algorithm which, given malgebra in a finite language, determines whether the variety generated by the algebra is congruence meet-semidistributive and residually less then m.