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 ...

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 . . . . . . . . . . . . . ....

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...

We develop a new p-adic algorithm to compute the minimal polynomial of a class invariant. Our approach works for virtually any modular function yielding class invariants. The main algorithmic tool is modular polynomials, a concept which we generalize to functions of higher level.

These groups are isomorphic to the free product of two finite cyclic groups of orders  and q. The first few Hecke groups are H(λ) = = PSL(,Z) (the modular group), H(λ) = H( √ ), H(λ) = H( + √   ), and H(λ) = H( √ ). It is clear from the above that H(λq) ⊂ PSL(,Z[λq]), but unlike in the modular group case (the case q = ), the inclusion is strict and the index [PSL(,Z[λq]) :H(λq)] i...

The mth Chebyshev polynomial of a square matrix A is the monic polynomial that minimizes the matrix 2-norm of p(A) over all monic polynomials p(z) of degree m. This polynomial is uniquely defined if m is less than the degree of the minimal polynomial of A. We study general properties of Chebyshev polynomials of matrices, which in some cases turn out to be generalizations of well known propertie...

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...