منابع مشابه
The Conjugate Dimension of Algebraic Numbers
We find sharp upper and lower bounds for the degree of an algebraic number in terms of the Q-dimension of the space spanned by its conjugates. For all but seven nonnegative integers n the largest degree of an algebraic number whose conjugates span a vector space of dimension n is equal to 2n!. The proof, which covers also the seven exceptional cases, uses a result of Feit on the maximal order o...
متن کاملDiophantine approximation by conjugate algebraic numbers
In 1969, Davenport and Schmidt provided upper bounds for the approximation of a real number by algebraic integers. Their novel approach was based on the geometry of numbers and involved the duality for convex bodies. In the present thesis we study the approximation of a real number by conjugate algebraic numbers. We find inspiration in Davenport and Schmidt’s method, but ultimately our approxim...
متن کاملConjugate Algebraic Numbers Close to a Symmetric Set
A new proof is presented for the Motzkin theorem saying that if a set consists of d− 1 complex points and is symmetric relative to the real axis, then there exists a monic, irreducible, and integral polynomial of degree d whose roots are as close to each of these d−1 points as we wish. Unlike the earlier proofs, the new proof is efficient, i.e., it gives both an explicit construction of the pol...
متن کاملRoots of Unity as Quotients of Two Conjugate Algebraic Numbers
Let α be an algebraic number of degree d > 2 over Q. Suppose for some pairwise coprime positive integers n1, . . . , nr we have deg(αj ) < d for j = 1, . . . , r, where deg(α) = d for each positive proper divisor n of nj . We prove that then φ(n1 . . . nr) 6 d, where φ stands for the Euler totient function. In particular, if nj = pj , j = 1, . . . , r, are any r distinct primes satisfying deg(α...
متن کاملEffective simultaneous approximation of complex numbers by conjugate algebraic integers
We study effectively the simultaneous approximation of n − 1 different complex numbers by conjugate algebraic integers of degree n over Z( √ −1). This is a refinement of a result of Motzkin [2] (see also [3], p. 50) who has no estimate for the remaining conjugate. If the n−1 different complex numbers lie symmetrically about the real axis, then Z( √ −1) can be replaced by Z. In Section 1 we prov...
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ژورنال
عنوان ژورنال: Annales Academiae Scientiarum Fennicae Series A I Mathematica
سال: 1975
ISSN: 0066-1953
DOI: 10.5186/aasfm.1975.582