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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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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 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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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 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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A principal ideal is an ideal in a ring that is generated by a single element, meaning every element of the ideal can be expressed as the ring element multiplied by any element of the ring. This concept is fundamental in ring theory as it simplifies the study of ideal structures, especially in commutative rings and polynomial rings.
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Unique factorization refers to the expression of a number or mathematical object as a product of prime elements in a way that is unique up to the order of factors. This property is fundamental in number theory and algebra, ensuring that every integer greater than 1 can be uniquely represented as a product of primes, known as the Fundamental Theorem of Arithmetic.
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The class number is an important invariant in algebraic number theory that measures the failure of unique factorization in the ring of integers of a number field. It provides insights into the structure of the ideal class group and has deep implications in the study of Diophantine equations and the distribution of prime numbers.
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Group theory is a branch of abstract algebra that studies the algebraic structures known as groups, which are sets equipped with an operation that satisfies four fundamental properties: closure, associativity, identity, and invertibility. It provides a unifying framework for understanding symmetry in mathematical objects and has applications across various fields including physics, chemistry, and computer science.
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A quotient group, also known as a factor group, is formed by partitioning a group into disjoint subsets called cosets of a normal subgroup, and the operation on these cosets mirrors the group operation. This construction is fundamental in group theory as it allows the analysis of group structures by simplifying them into smaller, more manageable pieces.
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Ideal theory is a methodological approach in political philosophy that focuses on envisioning a perfectly just society, often used as a standard to critique existing social structures. It contrasts with non-Ideal theory, which addresses the practical challenges and injustices prevalent in the real world.
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Class field theory is a major branch of algebraic number theory that describes the abelian extensions of a number field in terms of the field's arithmetic properties, particularly its ideal class group. It provides a profound connection between field theory and group theory, serving as a foundation for understanding more complex non-abelian extensions in modern number theory.
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The Hilbert class field of a number field is the maximal unramified abelian extension of that field, where every ideal becomes principal. It plays a crucial role in algebraic number theory as it connects the arithmetic of the number field with its ideal class group, providing insights into the structure of the field's extensions.
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Abelian extensions are a class of field extensions where the Galois group is an abelian group, meaning the group operation is commutative. These extensions are central to class field theory, which provides a comprehensive understanding of abelian extensions of number fields and their relation to ideal class groups and unit groups.
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Number fields are extensions of the rational numbers, characterized by the addition of roots of polynomials with rational coefficients. They serve as a fundamental structure in algebraic number theory, enabling the study of number properties through algebraic methods.
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Cyclotomic fields are number fields obtained by adjoining a primitive root of unity to the rational numbers, playing a crucial role in algebraic number theory and the study of Galois groups. They are instrumental in understanding the properties of integers and primes, particularly through their use in class field theory and the proof of Fermat's Last Theorem.
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An Abelian extension is a field extension whose Galois group is an Abelian group, meaning the group operation is commutative. These extensions are significant in number theory and algebra because they generalize the properties of cyclotomic fields and are central to class field theory, which describes the abelian extensions of a number field in terms of its ideal class group.
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The Artin map is a central concept in class field theory that establishes a connection between the ideal class group of a number field and the Galois group of its abelian extension. It provides a homomorphism that is pivotal in understanding the reciprocity laws governing the arithmetic of number fields.
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