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.
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.
An unramified extension is a field extension where no prime ideal of the base field's ring of integers becomes ramified in the extension. This means that every prime ideal in the base field's ring of integers either remains prime or splits into distinct prime ideals in the extension's ring of integers, preserving the notion of 'smoothness' in algebraic geometry and number theory.
A Principal Ideal Domain (PID) is an integral domain in which every ideal is generated by a single element, making it a generalization of the ring of integers. PIDs are important in algebraic number theory and algebraic geometry because they provide a framework for understanding divisibility and factorization in more complex rings.
Quadratic extensions are a type of field extension where the degree of the extension over its base field is two, typically formed by adjoining a square root of an element not already in the field. These extensions are fundamental in understanding the structure of algebraic numbers and play a crucial role in number theory and algebraic geometry.
A quadratic field is a type of number field obtained by adjoining the square root of a number, typically a non-square integer, to the field of rational numbers. This creates a field extension of degree two, which is fundamental in algebraic number theory and has applications in solving quadratic Diophantine equations and understanding the distribution of prime numbers.
Units in number fields are elements that have a multiplicative inverse within the field, forming a group under multiplication known as the unit group. Understanding the structure of this group is crucial for solving Diophantine equations and studying algebraic integers, as it reveals deep insights into the arithmetic properties of the field.
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.
An integrally closed domain is a commutative ring in which every element that is a root of a monic polynomial with coefficients from the ring is already an element of the ring itself. This property ensures that the ring is as 'complete' as possible with respect to the integral closure, meaning it contains all elements that should belong to it based on polynomial equations with coefficients in the ring.
A cyclotomic field is an extension of the rational numbers generated by a primitive root of unity, providing a rich structure for studying number theory and algebraic integers. These fields play a crucial role in solving classical problems like Fermat's Last Theorem and are deeply connected to Galois theory and modular forms.
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.
The norm of an element in mathematics, particularly in the context of algebraic number theory and field theory, is a function that maps elements of a field extension to the base field, providing a measure of the 'size' or 'magnitude' of the element. It plays a crucial role in understanding the structure of number fields, aiding in the classification of algebraic integers and the analysis of field extensions.
Dirichlet's Unit Theorem provides a fundamental insight into the structure of the unit group of the ring of integers in a number field, stating that this group is finitely generated and its rank is determined by the number of real and complex embeddings of the field. Essentially, it tells us that the units in such rings can be described by a finite basis, up to roots of unity, which is crucial for understanding the arithmetic of number fields.