By Jan-Hendrik Evertse, Kálmán Győry

ISBN-10: 1107097614

ISBN-13: 9781107097612

Discriminant equations are an enormous category of Diophantine equations with shut ties to algebraic quantity thought, Diophantine approximation and Diophantine geometry. This ebook is the 1st complete account of discriminant equations and their purposes. It brings jointly many features, together with powerful effects over quantity fields, potent effects over finitely generated domain names, estimates at the variety of strategies, purposes to algebraic integers of given discriminant, energy indispensable bases, canonical quantity structures, root separation of polynomials and aid of hyperelliptic curves. The authors' past name, Unit Equations in Diophantine quantity idea, laid the foundation through proposing very important effects which are used as instruments within the current booklet. This fabric is in short summarized within the introductory chapters in addition to the required uncomplicated algebra and algebraic quantity conception, making the e-book available to specialists and younger researchers alike.

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Extra resources for Discriminant Equations in Diophantine Number Theory

Example text

Further, let be a finite étale K-algebra with [ : K] = n. Since A is Noetherian, its integral closure A in is finitely generated as an A-module. Definition The discriminant ideal dA /A of A over A is defined as the ideal of A generated by all numbers D /K (α1 , . . , αn ) with α1 , . . , αn ∈ A . 1 below, which is formulated in a more general form for lattices, it follows that if G is any finite set of A-module generators of A , then dA /A is already generated by the numbers D /K (α1 , . . , αn ) with α1 , .

Every fractional ideal a of A can be extended to a fractional ideal aAL of AL ; this is the smallest fractional ideal of AL containing a. , 0 S , 1 ∈ S , and for all α, β ∈ S we have αβ ∈ S . Then S −1 A := {y−1 x : x ∈ A, y ∈ S } is an integral domain with quotient field K containing A, called the localization of A away from S . The elements of S are units of S −1 A. Every fractional ideal a of A can be extended to a fractional ideal S −1 a := {y−1 x : x ∈ a, y ∈ S } of S −1 A. 17 18 Dedekind Domains Definition Let A be an integral domain with quotient field K.

4. (ii)⇒(iii),(iv). Clearly, the coefficients of X /K;α are integral over A, and also, they belong to K. Hence they belong to A since A is integrally closed. 5. (iii),(iv)⇒(i). Clear, since α is a zero of both X /K;α and fα . The lemma clearly implies that Tr if α ∈ /K (α) ∈ A, N /K (α) ∈ A, D /K (α) ∈A is integral over A, and D if ω1 , . . , ωn ∈ /K (ω1 , . . , ωn ) = det Tr are integral over A. /K (ωi ω j ) 1≤i, j≤n ∈A 16 Finite Étale Algebras over Fields We keep our assumptions that A is an integrally closed integral domain with quotient field K of characteristic 0 and that is a finite étale K-algebra with [ : K] = n.

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Discriminant Equations in Diophantine Number Theory by Jan-Hendrik Evertse, Kálmán Győry

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