The genesis of ideal theory

Edwards, Harold M.

doi:10.1007/bf00327914

Cited by 58 publications

(25 citation statements)

References 7 publications

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“…Our work is complementary to existing constructive presentations of Riemann surfaces [19][20][21][22]. Our use of formal topology, which is a reformulation of Hilbert's notion of introduction and elimination of ideal elements, allows us to have access to the power of abstract methods (prime ideals, valuations), in the same way as [13,14] simplify some proofs of Bishop.…”

Section: Resultsmentioning

confidence: 97%

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Space of valuations

Coquand

2009

Annals of Pure and Applied Logic

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a b s t r a c t Formal topology presents a topological space, not as a set of points, but as a logical theory which describes the lattice of open sets. The application to Hilbert's program is then the following. Hilbert's ideal objects are represented by points of such a formal space. There are general methods to ''eliminate'' the use of points, close to the notion of forcing and to the ''elimination of choice sequences'' in intuitionist mathematics, which correspond to Hilbert's required elimination of ideal objects. This paper illustrates further this general program on the notion of valuations. They were introduced by Dedekind and Weber [R. Dedekind, H. Weber, Theorie des algebraischen Funktionen einer Veränderlichen, J. de Crelle t. XCII (1882) 181-290] to give a rigorous presentation of Riemann surfaces. It can be argued that it is one of the first example in mathematics of point-free representation of spaces [N. Bourbaki, Eléments de Mathématique. Algèbre commutative, Hermann, Paris, 1965, Chapitre 7]. It is thus of historical and conceptual interest to be able to represent this notion in formal topology.1 Logically, such a quantification is a priori a Π 1 1 statement and it is analyzed in the form of a Σ 0 1 equivalent assertion. 2 Technically, the introduction of a point of a formal space corresponds to working in the sheaf model over this space, and the elimination of this point is achieved by the Beth-Kripke-Joyal explanation of the logic of this sheaf model. In most cases, this elimination can be carried out directly without involving explicitly the notion of sheaf models. 0168-0072/$ -see front matterIf φ * has the going-up or going-down property and is surjective, it is clear in terms of points that this implies K dim Sp(Z ) K dim Sp(V ). The following proposition expresses this implication in a point-free way.Proposition 4. If φ : Z → V has the going-up or going-down property and is injective and K dim V < n then K dim Z < n.Proof. We give only the proof for the going-up property (the going-down property follows by duality). Let a 1 , . . . , a n be an arbitrary sequence in Z . Since K dim V < n we can find v 1 , . . . , v n in V such that Proposition 16. The spectrum of the lattice group Div(R) [13] is the dual of the Zariski spectrum of R.Proof. The spectrum of Div(R) is shown in [13] to be isomorphic to the lattice of positive elements of Div(R), that is the finitely generated ideal of R, with the order I J if and only if there exists n such that I J n . This is equivalent to saying that J is included into the radical of I.Proposition 17. If R is a Prüfer domain then the center map φ : Zar(R) → Val(R) is an isomorphism.7 In the localization R[1/u i ] the ideal x 1 , . . . , x n becomes principal and equal to x i . 8 Dedekind himself thought that the existence of such an inverse was the fundamental result about the ring of integers of an algebraic field of numbers [1]. Theorem 25 shows that this ring is a Prüfer domain.9 The structure of lattice group was discovered by Dedekind and rediscovered i...

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Section: Resultsmentioning

confidence: 97%

“…In this work with Weber [17], Dedekind used his newly created theory of ideals, a theory that has played an important rôle in the development of non-constructive methods in mathematics [19,21]. It is thus also relevant to illustrate Hilbert's notion of introduction and elimination of ideal elements in this context.…”

Section: Introductionmentioning

confidence: 99%

Space of valuations

Coquand

2009

Annals of Pure and Applied Logic

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“…Die Funktion ψr besteht aus ganzen positiven Zahlen, welche in die verschiedenen Potenzen von r multiplicirt sind, 14 Jacobi then continues: 12 Eventually, however, it turned out that the decomposition of Gauss sums only gives a piece of the reciprocity law for -th powers, namely Eisenstein's reciprocity law. This is enough to derive the full version for cubic and quartic residues, but for higher powers, Kummer had to generalize Gauss's genus theory from quadratic forms to class groups in Kummer extensions of cyclotomic number fields.…”

Section: Jacobi Mapsmentioning

confidence: 98%

“…13 Cauchy also studied these sums, but his lack of understanding higher reciprocity kept him from going as far as Jacobi did. 14 The function ψr consists of positive integers, multiplied by the different powers of r.…”

Section: Jacobi Mapsmentioning

confidence: 99%

Jacobi and Kummer’s ideal numbers

Lemmermeyer¹

2009

Abh. Math. Semin. Univ. Hambg.

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“…Learning this standard definition a student may wonder where the funny name "ideal" comes from. The definition itself gives no clue to it but historical works provide a clear answer (see Edwards 1980). The term stems from the notion of ideal number (Idealzahl) introduced by Kummer in 1847.…”

Section: Lost Idealsmentioning

confidence: 99%

How Mathematical Concepts Get Their Bodies

Rodin

2010

Topoi

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The genesis of ideal theory

Cited by 58 publications

References 7 publications

Space of valuations

Space of valuations

Jacobi and Kummer’s ideal numbers

How Mathematical Concepts Get Their Bodies

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