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Algorithmic Breakthroughs in Factoring Sparse Polynomials over Algebraic Number Fields
SÉRGIO DE ANDRADE, PAULO
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A novel color images security-based on SPN over the residue classes of quaternion integers [Formula: see text]. [PDF]
Sci RepSajjad M, Alqwaifly NA.
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Realizing Algebraic Number Fields
1983In the paper [13], the authors studied the problem of realizing rational division algebras in a special way. Let D be a division algebra that is finite dimensional over the rational field Q. If p is a prime, we say that D is p-realizable when there is a p-local torsion free abelian group A whose rank is the dimension of D over Q, such that D is ...
R. S. Pierce, C. I. Vinsonhaler
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ON NONMONOGENIC ALGEBRAIC NUMBER FIELDS
Rocky Mountain Journal of Mathematics, 2023Let \(p\) be a prime number and \(f (x) = x^{p^ s}- ax^m- b\) belonging to \(\mathbb Z[x]\) be an irreducible polynomial. Let \( K = \mathbb Q(\theta )\) be an algebraic number field with \(\theta\) a root of \( f (x)\). Let \(r_1\) stand for the highest power of \(p\) dividing \(b^{p^s}- b -ab^m.\) This paper gives some explicit conditions involving ...
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2016
Arithmetical algorithms considered in Chap. 5 are based on the arithmetical operations with matrices of the number systems. If the entries of these matrices are not integers or rationals, we need arithmetical algorithms which work with them. Such algorithms exist for algebraic numbers.
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Arithmetical algorithms considered in Chap. 5 are based on the arithmetical operations with matrices of the number systems. If the entries of these matrices are not integers or rationals, we need arithmetical algorithms which work with them. Such algorithms exist for algebraic numbers.
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2019
A complex number \(\xi \) is called an algebraic integer if \(\mathbf {Z}[ \xi ]\) is a finitely generated \(\mathbf {Z}\)-module; this condition is equivalent to the fact that \(f( \xi )=0\) for some polynomial \(f(X)=X^m+a_1X^{m-1}+ \cdots +a_m\), \(a_i \in \mathbf {Z}\). Let \(\mathbf {A}\) be the set of all algebraic integers.
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A complex number \(\xi \) is called an algebraic integer if \(\mathbf {Z}[ \xi ]\) is a finitely generated \(\mathbf {Z}\)-module; this condition is equivalent to the fact that \(f( \xi )=0\) for some polynomial \(f(X)=X^m+a_1X^{m-1}+ \cdots +a_m\), \(a_i \in \mathbf {Z}\). Let \(\mathbf {A}\) be the set of all algebraic integers.
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Journal of Soviet Mathematics, 1987
Translation from Itogi Nauki Tekh., Ser. Algebra Topol. Geom. 22, 117--204 (Russian) (1984; Zbl 0563.12002).
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Translation from Itogi Nauki Tekh., Ser. Algebra Topol. Geom. 22, 117--204 (Russian) (1984; Zbl 0563.12002).
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1967
We shall need some elementary results about vector-spaces over Q, involving the following concept: Definition 1. Let E be a vector-space of finite dimension over Q. By a Q-lattice in E, we understand a finitely generated subgroup of E which contains a basis of E over Q. Proposition 1.
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We shall need some elementary results about vector-spaces over Q, involving the following concept: Definition 1. Let E be a vector-space of finite dimension over Q. By a Q-lattice in E, we understand a finitely generated subgroup of E which contains a basis of E over Q. Proposition 1.
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1979
Any nonconstant polynomial with rational coefficients has roots in the complex numbers. Those complex numbers which are roots of polynomials with rational coefficients are called algebraic numbers.
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Any nonconstant polynomial with rational coefficients has roots in the complex numbers. Those complex numbers which are roots of polynomials with rational coefficients are called algebraic numbers.
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