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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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On the units of algebraic number fields
Mathematika, 1967Let p be a prime number, Qp the field of p-adic numbers and Ωp the completion of the algebraic closure of Qp with its valuation normed by setting |p| = 1/p.
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Arithmetic in an Algebraic Number Field
20106.1. Let F be a field. A map \(\,\nu : F\to {\bold{R}}\cup\{\infty\}\) is called an order function of F if it satisfies the following conditions:
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Groups of Algebras Over an Algebraic Number Field
American Journal of Mathematics, 1943Saunders MacLane, O. F. G. Schilling
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