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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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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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ALGEBRAIC NUMBER FIELDS WITH LARGE CLASS NUMBER
Mathematics of the USSR-Izvestiya, 1974zbMATH Open Web Interface contents unavailable due to conflicting licenses.
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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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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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Groups of Algebras Over an Algebraic Number Field
American Journal of Mathematics, 1943MacLane, S., Schilling, O. F. G.
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C*-algebraic approach to interacting quantum field theory: inclusion of Fermi fields
Letters in Mathematical Physics, 2022Romeo Brunetti +2 more
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Intuitionistic Fuzzy Algebraic Field Extensions
Lecture Notes in Networks and Systems, 2023Said Melliani, M Elomari
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