Let K be a field of characteristic not p (an odd prime), containing a primitive p-th root of unity ζ, and let L = K[z] with x n − a the minimal polynomial of z over K: thus L|K is a Kummer extension, with cyclic Galois group G = 〈σ〉 acting on L via σ(z) = ζz. T. Kohl, 1998, showed that L|K has pn−1 Hopf Galois structures. In this paper we describe these Hopf Galois structures.

Let K ≤ F be a normal field extension. Put S := S(K,F) and P := P(K,F). Let K ≤ E ≤ S. (a) We will show that S(E,F) = S. Let a ∈ S. Then a is separable over K. By 5.2.20, a is also separable over E, so a ∈ S(E,F) and S ⊆ S(E,F). Now note that E is a separable extension of K because E ≤ S. Let a ∈ S(E,F). Since a is separable over E, E(a) is a separable extension of E. Thus, K ≤ E ≤ E(a) is a se...

We find all 15909 algebraic integers whose conjugates all lie in an ellipse with two of them nonreal, while the others lie in the real interval [−1, 2]. This problem has applications to finding certain subgroups of SL(2,C). We use explicit auxiliary functions related to the generalized integer transfinite diameter of compact subsets of C. This gives good bounds for the coefficients of the minim...

We show that the number α = (1 + √ 3 + 2 √ 5)/2 with minimal polynomial x4 − 2x3 + x − 1 is the only Pisot number whose four distinct conjugates α1, α2, α3, α4 satisfy the additive relation α1+α2 = α3+α4. This implies that there exists no two non-real conjugates of a Pisot number with the same imaginary part and also that at most two conjugates of a Pisot number can have the same real part. On ...

Let L = Q[α] be an abelian number field of prime degree q, and let a be a nonzero rational number. We describe an algorithm which takes as input a and the minimal polynomial of α over Q, and determines if a is a norm of an element of L. We show that, if we ignore the time needed to obtain a complete factorization of a and a complete factorization of the discriminant of α, then the algorithm run...

Recently, motivated by the study of vectorized stream cipher systems, the joint linear complexity and joint minimal polynomial of multisequences have been investigated. Let S be a linear recurring sequence over finite field Fqm with minimal polynomial h(x) over Fqm. Since Fqm and F m q are isomorphic vector spaces over the finite field Fq, S is identified with an m-fold multisequence S over the...

An algorithm which either finds an nonzero integer vector m for given t real n-dimensional vectors x1, · · · , xt such that xi m = 0 or proves that no such integer vector with norm less than a given bound exists is presented in this paper. The cost of the algorithm is at most O(n4 + n3 log λ(X)) exact arithmetic operations in dimension n and the least Euclidean norm λ(X) of such integer vectors...

An 1861 theorem of Ch. Hermite [He] asserts that every field extension (and more generally, every étale algebra) E/F of degree 5 can be generated by an element a ∈ E whose minimal polynomial is of the form f(x) = x + b2x 3 + b4x+ b5 . Equivalently, trE/F (a) = trE/F (a ) = 0. A similar result for étale algebras of degree 6 was proved by P. Joubert in 1867; see [Jo]. It is natural to ask whether...

Let L = K() be an abelian extension of degree n of a number eld K, given by the minimal polynomial of over K. We describe an algorithm for computing the local Artin map associated to the extension L=K at a nite or innnite prime v of K. We apply this algorithm to decide if a nonzero a 2 K is a norm from L, assuming that L=K is cyclic.

Generalized GIPSCAL, like DEDICOM, is a model for the analysis of square asymmetric tables. It is a special case of DEDICOM, but unlike DEDICOM, it ensures the nonnegative definiteness (nnd) of the model matrix, thereby allowing a spatial representation of the asymmetric relationships among “objects”. A fast convergent algorithm was developed for GIPSCAL with acceleration by the minimal polynom...