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A fractional ideal is a generalization of the concept of an ideal in a ring, allowing for denominators from the ring itself, and is crucial in the study of Dedekind domains where every fractional ideal can be uniquely factored into prime ideals. This concept is particularly important in algebraic number theory, as it helps in understanding the structure of rings of integers in number fields and their divisibility properties.
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Ring theory is a branch of abstract algebra that studies rings, which are algebraic structures consisting of a set equipped with two binary operations that generalize the arithmetic of integers. It is fundamental in understanding structures such as fields, modules, and algebras, and has applications in number theory, geometry, and physics.
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The concept of 'Ideal' often represents a standard of perfection or excellence that serves as a goal or model to be strived for, yet it is frequently unattainable in reality. It plays a crucial role in philosophical, ethical, and social contexts, guiding behavior, aspirations, and evaluations of value and worth.
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A Dedekind domain is an integral domain in which every nonzero proper ideal factors uniquely into a product of prime ideals, making it a generalization of the ring of integers. It plays a critical role in algebraic number theory and algebraic geometry due to its ideal-theoretic properties and connections to Noetherian rings and divisors.
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Algebraic number theory is a branch of number theory that studies the properties of algebraic numbers, which are roots of non-zero polynomial equations with rational coefficients. It connects number theory with abstract algebra, particularly through the use of field theory, Galois theory, and ring theory to solve problems related to integers and their generalizations.
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A prime ideal in a ring is an ideal whose complement is multiplicatively closed, meaning if a product of two elements is in the ideal, then at least one of the elements is in the ideal. Prime ideals are fundamental in algebraic geometry and commutative algebra as they generalize the notion of prime numbers to more abstract algebraic structures, serving as building blocks for the structure of rings.
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The ring of integers is a fundamental structure in algebraic number theory, consisting of all algebraic integers within a given number field, forming a ring under addition and multiplication. It generalizes the concept of integers to more complex number systems, providing a framework for understanding divisibility, factorization, and the arithmetic properties of algebraic numbers.
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A number field is a finite degree field extension of the rational numbers, providing a rich structure for studying algebraic numbers and their properties. Number fields are fundamental in algebraic number theory, as they generalize the rational numbers and allow for the exploration of Diophantine equations and prime factorization in more complex settings.
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Divisibility is a fundamental concept in number theory that determines whether one integer can be divided by another without leaving a remainder. Understanding divisibility helps in simplifying fractions, finding greatest common divisors, and solving problems involving modular arithmetic.
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A Dedekind domain is an integral domain in which every nonzero proper ideal factors uniquely into prime ideals, making it a generalization of the ring of integers in number fields. These domains play a crucial role in algebraic number theory and algebraic geometry due to their ideal-theoretic properties and the structure they provide in studying rings and fields.
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The ideal class group is a fundamental invariant in algebraic number theory that measures the failure of unique factorization in the ring of integers of a number field. It is defined as the quotient of the group of fractional ideals by the subgroup of principal ideals, providing insight into the arithmetic structure of the field.
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