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On Validated Computing in Algebraic Number Fields [PDF]
Journal of Symbolic Computation, 1997 The author considers the use of decimal approximations (rather than integer arithmetic) in number field computations required (say) for finding units, e.g., divisibility of two numbers in an order and finding numbers with given norm. The LLL-algorithm is useful in ensuring scaled matrices for the basis (and conjugates).
MICHAEL E. POHST, POHST, MICHAEL E.
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Isomorphisms of algebraic number fields [PDF]
Journal de théorie des nombres de Bordeaux, 2012 Let ℚ ( α ) and ℚ (
Mark van Hoeij, Vivek Pal
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On the Units of Algebraic Number Fields
Journal of Number Theory, 1994 Let \(K\) be an algebraic number field of degree \(n\) over the rational number field and \(k\) be a proper subfield of \(K\) with \([K: k]=m\). Further, let \(R_ 1\), \(r_ 1\) denote the number of embeddings of \(K\), \(k\) into the real numbers and \(2R_ 2\), \(2r_ 2\) denote the numbers of embeddings of \(K\), \(k\) into the complex numbers.
Yamaguchi, I., Takeuchi, H.
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The Genus Field and Genus Number in Algebraic Number Fields [PDF]
Nagoya Mathematical Journal, 1967 Let k be an algebraic number field and K be its normal extension of finite degree. Then the genus field K* of K over k is defined as the maximal unramified extension of K which is obtained from K by composing an abelian extension over k2). We call the degree (K*: K) the genus number of K over k.
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On the Genus Field in Algebraic Number Fields
Tokyo Journal of Mathematics, 1983 Sei \(K/k\) eine endliche galoissche Erweiterung algebraischer Zahlkörper, \(\mathfrak M\) ein Modul von \(K\) (der unendliche Primstellen enthalten kann), \(K(\mathfrak M)\) der Strahlklassenkörper von \(K \bmod {\mathfrak M}\), \(E/k\) die maximale abelsche Erweiterung in \(K(\mathfrak M)\) und \(K^*(\mathfrak M) = E\cdot K\); \(K^*(\mathfrak M ...
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Computing in the Field of Complex Algebraic Numbers
Journal of Symbolic Computation, 1997 zbMATH Open Web Interface contents unavailable due to conflicting licenses.
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The geometry of numbers over algebraic number fields [PDF]
Transactions of the American Mathematical Society, 1958 1. The Geometry of Numbers was founded by Minkowski in order to attack certain arithmetical problems, and is normally concerned with lattices over the rational integers. Minkowski himself, however, also treated a special problem over complex quadratic number fields [5], and a number of writers have since followed him.
Rogers, K., Swinnerton-Dyer, H. P. F.
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On the abc$abc$ conjecture in algebraic number fields
Mathematika, 2023 AbstractIn this paper, we prove a weak form of the conjecture generalised to algebraic number fields. Given integers satisfying , Stewart and Yu were able to give an exponential bound in terms of the radical over the integers (Stewart and Yu [Math. Ann. 291 (1991), 225–230], Stewart and Yu [Duke Math. J. 108 (2001), no. 1, 169–181]), whereas Győry was
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Algebraic leaves of algebraic foliations over number fields [PDF]
Publications Mathématiques de l'IHÉS, 2001 We prove an algebraicity criterion for leaves of algebraic foliations defined over number fields. Namely, consider a number field K embedded in C , a smooth ...
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Normal Algebraic Number Fields [PDF]
Proceedings of the National Academy of Sciences, 1940 MacLane, Saunders, Schilling, O. F. G.
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