An order bounded disjointness preserving operator T on an Archimedean vector lattice is algebraic if and only if the restriction of Tn! to the vector sublattice generated by the range of Tm is strongly diagonal, where n is the degree of the minimal polynomial of T and m is its ‘valuation’.

Let L/K be a separable field extension of degree 6. An 1867 theorem of P. Joubert asserts that if char(K) 6= 2 then L is generated over K by an element whose minimal polynomial is of the form t + at + bt + ct+ d for some a, b, c, d ∈ K. We show that this theorem fails in

This lecture introduces the Jordan canonical form of a matrix — we prove that every square matrix is equivalent to a (essentially) unique Jordan matrix and we give a method to derive the latter. We also introduce the notion of minimal polynomial and we point out how to obtain it from the Jordan canonical form. Finally, we make an encounter with companion matrices. 1 Jordan form and an applicati...

8 Matrices 16 8.1 Representation of vector spaces and linear transformations . . . . . . . . . . . . . . . 16 8.2 Similarity transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 8.3 Direct sums of matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 8.4 The companion matrix of a polynomial . . . . . . . . . . . . . . . . . . . . . . . ....

With the exception of a nite set of nite diierential Galois groups, if an irreducible linear diierential equation L(y) = 0 of prime order with unimodular diierential Galois group has a Liouvillian solution, then all algebraic solutions of smallest degree of the associated Riccati equation are solutions of a unique minimal polynomial. If the coeecients of L(y) = 0 are in Q()(x) Q(x) this unique ...

4 The minimal polynomial of a linear transformation 7 4.1 Existence of the minimal polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 4.2 The minimal polynomial for algebraically closed fields . . . . . . . . . . . . . . . . . 8 4.3 The characteristic polynomial and the Cayley–Hamilton theorem . . . . . . . . . . . 9 4.4 Finding the minimal polynomial . . . . . . . . . . . . . ....

We propose a new computation method for the linear complexity and the minimal polynomial of Ding-Helleseth-generalized cyclotomic sequences. We will find the linear complexity of DingHelleseth-generalized cyclotomic sequences of order four and six and make the results of Tongjiang Yan et. al [19] about the sequences of order four more specific. 2010 Mathematics Subject Classifications: 11B50, 9...

In this paper we consider the following problem. Let k < K be fields. Suppose S ⊂ Mn(K) is an absolutely irreducible multiplicative semigroup with the property that the spectrum of every a ∈ S is contained in k. Does it follow that S is simultaneously realizable in Mn(k), that is, does there exist a p ∈ GLn(K) such that pSp−1 ⊂ Mn(k)? The overfield K plays no essential role in this situation. C...

The paper arose from the fact that the smallest element of the set of Salem numbers is not known. Indeed, it is not even known whether this set has a smallest element. The aim of this paper is to prove that the minimal polynomial of the smallest Salem series of degree n in the field of formal power series over a finite field is given by P (Y ) = Y n −XY n−1 − Y + X − 1, where we suppose that 1 ...