5 edition of **Diophantine Equations and Power Integral Bases in Algebraic Number fields** found in the catalog.

- 170 Want to read
- 13 Currently reading

Published
**April 26, 2002**
by Birkhäuser Boston
.

Written in English

- Algebraic Number Theory,
- Calculus & mathematical analysis,
- Group Theory,
- Mathematics,
- Science/Mathematics,
- Programming - Algorithms,
- Computer Science,
- Number Theory,
- Algorithmic Analysis,
- Algorithms,
- Mathematics / General,
- Mathematics of Computing,
- Bases (Linear topological spac,
- Algebraic Fields,
- Bases (Linear topological spaces),
- Diophantine equations

The Physical Object | |
---|---|

Format | Paperback |

Number of Pages | 208 |

ID Numbers | |

Open Library | OL8074765M |

ISBN 10 | 0817642714 |

ISBN 10 | 9780817642716 |

In mathematics, a monogenic field is an algebraic number field K for which there exists an element a such that the ring of integers O K is the subring Z[a] of K generated by O K is a quotient of the polynomial ring Z[X] and the powers of a constitute a power integral basis.. In a monogenic field K, the field discriminant of K is equal to the discriminant of the minimal . Mathematical Reviews number (MathSciNet) MR Zentralblatt MATH identifier Subjects Primary: 11D Multiplicative and norm form equations Secondary: 11Y Computer solution of Diophantine equations. Keywords Power integral bases simplest quartic fields index form equations. Citation.

Stack Exchange network consists of Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.. Visit Stack Exchange. I. Gaál, Diophantine equations and power integral bases: new computational methods, Birkhäuser, Mathematical Reviews (MathSciNet): MR I. Gaál and B. Jadrijević, “Determining elements of minimal index in an infinite family of totally real bicyclic biquadratic number fields”, JP J. Algebra Number Theory Appl. (

research topics: diophantine equations, Thue equation, monogenity of number fields, power integral bases infnite parametric families of number fields; author of about 75 paper and a book on power integral basis; Kratki životopis – László Remete: mathematics BSc and Msc studies at University of Debrecen; from PhD student. algebraic number field applied arise Cassels Chapter class number congruence mod cubic curve cubic equation cubic field degree diophantine equation equation 19 equation ax equation f(x exist Fermat finite number finite set fundamental solution fundamental unit gives Hence homogeneous impossible infinity of integer infinity of solutions integer.

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Diophantine Equations and Power Integral Bases will be ideal for graduate students and researchers interested in the area.A basic understanding of number fields and algebraic methods to solve Diophantine equations is required.

This monograph investigates algorithms for determining power integral bases in algebraic number fields. It introduces the best-known methods for solving several types of diophantine equations using Baker-type estimates, reduction methods, and enumeration algorithms.

Particular emphasis is placed on properties of number fields and new applications. The text is illustrated with several tables of various number fields, including their data on power integral bases.; Several interesting properties of number fields are examined.; Some infinite parametric families of fields are also considered as well as the resolution of the corresponding infinite parametric families of diophantine by: Diophantine Equations and Power Integral Bases will be ideal for graduate students and researchers interested in the area.

A basic understanding of number fields and algebraic methods to solve Diophantine equations is required. The existence of power integral bases in algebraic number fields is a classical problem in algebraic number theory [4, 6, 11].

It is especially delicate Author: István Gaál. Pethő, A. (), Connections between power integral bases and radix representationsin algebraic number fields, in: Yokoi-Chowla Conjecture and Related Problems, Furukawa Total Printing Co.

LTD, Saga, Japan. – Unique bookclosest competitor, Smart, Cambridge, does not treat index form equations. Author is a leading researcher in the field of computational algebraic number theory. The text is illustrated with several tables of various number fields, including their data on power integral bases.

Several interesting properties of number fields are. Computing power integral bases in algebraic number fields II. Algebraic Number Theory and Diophantine Analysis: Proceedings of the International Conference held in Graz, Austria, August 30 to September 5, (pp.

The authors' previous title, Unit Equations in Diophantine Number Theory, laid the groundwork by presenting important results that are used as tools in the present book.

This material is briefly summarized in the introductory chapters along with the necessary basic algebra and algebraic number theory, making the book accessible to experts and. This book is the first comprehensive account of discriminant equations and their applications.

It brings together many aspects, including effective results over number fields, effective results over finitely generated domains, estimates on the number of solutions, applications to algebraic integers of given discriminant, power integral bases. [13] GAÁL, I.-REMETE, L.: Binomial Thue equations and power integral bases in pure quartic fields, JP J.

Algebra Number Theory Appl. 32 (), [14] GAÁL, I.-SZABÓ, T.: Relative power integral bases in infinite families of quartic extensions of quadratic field, JP J. Algebra Number Theory Appl. 29 (), Diophantine geometry and, more generally, arithmetic geometry is the study of the points of an algebraic variety with coordinates in fields that are not algebraically closed and occur in algebraic number theory, such as the field of rational numbers, number fields, finite fields, function fields, and p-adic fields.

A Hasse-type principle for exponential Diophantine equations over number fields and its applications. Monatshefte für Mathematik, Vol.Issue.

3, p. 'Understanding the book requires only basic knowledge in algebra (groups, commutative rings, fields, Galois theory and elementary algebraic number theory). and J. P., Bell. This monograph investigates algorithms for determining power integral bases in algebraic number fields.

It introduces the best-known methods for solving several types of diophantine equations using Baker-type estimates, reduction methods, and enumeration algorithms.

Particular emphasis is placed on properties of number fields and new applications. The text is. Discriminant equations are an important class of Diophantine equations with close ties to algebraic number theory, Diophantine approximation and Diophantine geometry. This book is the first comprehensive account of discriminant equations and their applications.

It brings together many aspects, including effective results over number fields, effective results over. Diophantine equations can be reduced modulo primes, and then occur in coding theory and cryprography.

For example elliptic curve cryptography is based on doing calculations in finite field. The problem of determining power integral bases in algebraic number fields is equivalent to solving the corresponding index form equations.

As is known (cf. Gyory [25]), every index form equation can be reduced to an equation system consisting of unit equations in two variables over the normal closure of the original field. Get this from a library. Diophantine equations and power integral bases: new computational methods.

[István Gaál] -- "Advanced undergraduates and graduates will benefit from this exposition of methods for solving some classical types of diophantine equations. Researchers in the field will find new applications for.

Comments. The most outstanding recent result in the study of Diophantine equations was the proof by G. Falting of the Mordell conjecture, stating that curves of genus $ > 1 $(cf. Genus of a curve) over algebraic fields have no more than a finite number of rational points (cf.).From this result it follows, in particular, that the Fermat equation $ x ^ {n} + y ^ {n}.

Diophantine equations 11D57 Multiplicative and norm form equations Algebraic number theory: global fields 11R04 Algebraic numbers; rings of algebraic integers 11R16 Cubic and quartic extensions Computational number theory 11Y50 Computer solution of Diophantine equations.

Computing All Power Integral Bases of Cubic Fields By I. Gaál* and N. Schulte Abstract. Applying Baker's effective method and the reduction procedure of Baker and Davenport, we present several lists of solutions of index form equations in (totally real and complex) cubic algebraic number fields.Neither does a general decision procedure exist for finding solutions of systems of such Diophantine equations in the ring of integers of any number field of finite degree over $ \mathbf Q $.

Let $$ \tag{a1 } f _ {i} (X _ {1} \dots X _ {n}) = 0, 1 \leq i \leq r, $$ be a system of polynomial equations having integral coefficients.

MA Algebra and Number Theory Important Questions and Lecture Notes Properties - Homomorphism - Isomorphism - Cyclic groups - Cosets - Lagrange's theorem. Rings: Definition - Sub rings - Integral domain - Field - Integer modulo n - Ring homomorphism.

UNIT II FINITE FIELDS AND POLYNOMIALS Linear Diophantine equations – Congruence.