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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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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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2012
This chapter is an introduction to algebraic number fields, which arose from both a generalization of the arithmetic in ℤ and the necessity to solve certain Diophantine equations. After recalling basic concepts from algebra and providing some polynomial irreducibility tools, the ring of integers \(\mathcal {O}_{\mathbb {K}}\) of an algebraic number ...
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This chapter is an introduction to algebraic number fields, which arose from both a generalization of the arithmetic in ℤ and the necessity to solve certain Diophantine equations. After recalling basic concepts from algebra and providing some polynomial irreducibility tools, the ring of integers \(\mathcal {O}_{\mathbb {K}}\) of an algebraic number ...
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1981
A system of complex numbers is called a number field (or, more briefly, a field) if it contains more than one number and if along with the numbers α and β it always contains α + β, α − β,αβ, and, if β ≠ 0, α/β.
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A system of complex numbers is called a number field (or, more briefly, a field) if it contains more than one number and if along with the numbers α and β it always contains α + β, α − β,αβ, and, if β ≠ 0, α/β.
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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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1988
Let us consider the diophantine equation $$x^2 - dy^2 = 1$$ (4.1) , erroneously called Pell’s equation. (For its history, see Ref. 9.) Here d ≠ 0 is a square-free integer. We seek the integer solutions of (4.1). If d 1, it is a nontrivial fact that (4.1) has infinitely many solutions in integers.
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Let us consider the diophantine equation $$x^2 - dy^2 = 1$$ (4.1) , erroneously called Pell’s equation. (For its history, see Ref. 9.) Here d ≠ 0 is a square-free integer. We seek the integer solutions of (4.1). If d 1, it is a nontrivial fact that (4.1) has infinitely many solutions in integers.
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1993
An algebraic number field F is a finite extension field of the rational numbers ℚ. It can be generated by a root p of a monic irreducible polynomial $$f(t) = {{t}^{n}} + {{a}_{1}}{{t}^{{n - 1}}} + {\text{ }} \ldots + {{a}_{n}}\epsilon \mathbb{Z}[t]$$ , (27) where n is also called the degree of F.
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An algebraic number field F is a finite extension field of the rational numbers ℚ. It can be generated by a root p of a monic irreducible polynomial $$f(t) = {{t}^{n}} + {{a}_{1}}{{t}^{{n - 1}}} + {\text{ }} \ldots + {{a}_{n}}\epsilon \mathbb{Z}[t]$$ , (27) where n is also called the degree of F.
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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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Polynomial Computability of Fields of Algebraic Numbers
Доклады академии наук, 2018© 2018, Pleiades Publishing, Ltd. We prove that the field of complex algebraic numbers and the ordered field of real algebraic numbers have isomorphic presentations computable in polynomial time. For these presentations, new algorithms are found for evaluation of polynomials and solving equations of one unknown.
V. L. Selivanov +3 more
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Algebraic Numbers and Number Fields
1998A number α is called an algebraic number if it satisfies an equation of degree m of the form $${\alpha ^m} + {a_1}{\alpha ^{m - 1}} + {a_2}{\alpha ^{m - 2}} + \cdots + {a_m} = 0$$ where a 1, a 2,..., a m are rational numbers.
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