10.2 Cyclotomic fields and cyclotomic polynomials
4 min read•july 30, 2024
Cyclotomic fields are extensions of rational numbers that include roots of unity. They're key players in number theory, helping us tackle tricky equations and uncover hidden patterns in numbers. Their structure and properties make them super useful in various math fields.
Cyclotomic polynomials are the building blocks of these fields. They're special polynomials with whole number coefficients that define the roots of unity. These polynomials pop up everywhere, from solving ancient math puzzles to modern-day computer security.
Cyclotomic Fields and Properties
Definition and Basic Structure
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- Cyclotomic fields form by adjoining roots of unity to rational numbers Q
- Nth Q(ζn) created by adding primitive nth root of unity ζn to Q
- Degree of nth cyclotomic field over Q equals φ(n) (φ represents Euler's totient function)
- Abelian extensions of Q with over Q abelian
- Z[ζn] in Q(ζn) forms a Dedekind domain
- Play crucial role in number theory (study of Diophantine equations, reciprocity laws)
- Discriminant of Q(ζn) calculated as (product over prime divisors p of n)
Applications and Significance
- Fundamental in solving Diophantine equations (equations with integer solutions)
- Provide framework for studying reciprocity laws (relationships between different number fields)
- Used in cryptography (RSA algorithm relies on properties of cyclotomic polynomials)
- Important in algebraic number theory (studying properties of algebraic integers)
- Applied in coding theory (constructing error-correcting codes)
- Utilized in harmonic analysis (Fourier transforms related to cyclotomic fields)
Constructing Cyclotomic Polynomials
Definition and Construction
- Nth Φn(x) serves as of primitive nth root of unity over Q
- Recursive construction using formula (d runs through positive divisors of n)
- Degree of Φn(x) equals φ(n) (φ represents Euler's totient function)
- Monic polynomials with integer coefficients
- Irreducible over Q for all positive integers n (fundamental property)
- Galois group of Φn(x) over Q isomorphic to (Z/nZ)* (multiplicative group of units modulo n)
- Special case for prime p:
Properties and Examples
- Coefficients of cyclotomic polynomials always integers (not obvious from definition)
- First few cyclotomic polynomials:
- Möbius inversion formula used to express Φn(x) explicitly
- Cyclotomic polynomials satisfy various identities ( for odd n)
- Coefficients of Φn(x) not always ±1 (first occurrence for n = 105)
- Used in various mathematical applications (coding theory, cryptography)
Galois Groups of Cyclotomic Fields
Structure and Properties
- Galois group Gal(Q(ζn)/Q) isomorphic to (Z/nZ)*
- Automorphisms defined as σa: ζn → ζn^a for a coprime to n
- Order of Galois group equals φ(n) (reflecting degree of extension)
- Subfields of Q(ζn) correspond to subgroups of (Z/nZ)* via Galois correspondence
- For divisor d of n, Q(ζd) forms subfield of Q(ζn)
- Degree of extension [Q(ζn):Q(ζd)] calculated as φ(n)/φ(d)
- Maximal real subfield Q(ζn + ζn^-1) has index 2 in Q(ζn) for n > 2
Subfields and Compositums
- Compositum of cyclotomic fields Q(ζm) and Q(ζn) equals Q(ζlcm(m,n))
- Fixed field of subgroup {±1} of (Z/nZ)* forms maximal totally real subfield of Q(ζn)
- Intermediate fields between Q and Q(ζn) correspond to subgroups of (Z/nZ)*
- Galois group of Q(ζn) over Q(ζm) isomorphic to kernel of natural map (Z/nZ)* → (Z/mZ)*
- Intersection of Q(ζm) and Q(ζn) equals Q(ζgcd(m,n))
- Norm of cyclotomic units generates important subgroup of units in Q(ζn)
Cyclotomic Fields and Fermat's Last Theorem
Historical Approaches
- Cyclotomic fields crucial in proving Fermat's Last Theorem (FLT)
- Kummer's approach studied of x^p + y^p in Z[ζp] for prime p
- Regular primes (defined by divisibility properties in cyclotomic fields) central to Kummer's work
- Iwasawa theory examines behavior of ideal class groups in cyclotomic field towers
- Frey curve (elliptic curve from hypothetical FLT solution) analyzed using cyclotomic field properties
- Modularity of elliptic curves over Q connected to Galois representations from cyclotomic fields
Modern Developments
- Cyclotomic units and ideal class groups in cyclotomic fields contribute to understanding FLT
- Wiles' proof of FLT heavily relies on properties of cyclotomic fields
- Iwasawa's Main Conjecture (relating p-adic L-functions to ideal class groups) important in FLT study
- Cyclotomic fields used in studying generalizations of FLT (abc conjecture, Catalan's conjecture)
- Connections between cyclotomic fields and elliptic curves crucial in modern number theory
- Study of cyclotomic fields led to development of broader class field theory
Key Terms to Review (18)
Abelian extension: An abelian extension is a field extension of a number field that is both normal and separable, where the Galois group of the extension is an abelian group. This concept plays a crucial role in understanding the relationships between number fields and their arithmetic properties, linking to various advanced topics like cyclotomic fields and class field theory, which address the structure of these extensions and their applications in algebraic number theory.
Carl Friedrich Gauss: Carl Friedrich Gauss was a prominent German mathematician and scientist who made significant contributions to various fields, including number theory, statistics, and algebra. His work laid the groundwork for modern number theory, influencing concepts such as unique factorization and the study of integer solutions, while also advancing mathematical techniques that are essential in understanding discriminants and cyclotomic fields.
Cyclotomic field: A cyclotomic field is a type of number field obtained by adjoining a primitive $n$th root of unity, denoted as $\zeta_n$, to the field of rational numbers $\mathbb{Q}$. These fields play a significant role in number theory as they help in understanding the structure of extensions of $\mathbb{Q}$, particularly through the lens of Galois theory and the properties of integral bases.
Cyclotomic polynomial: A cyclotomic polynomial is a special type of polynomial defined as the product of linear factors corresponding to the primitive roots of unity. Specifically, the nth cyclotomic polynomial, denoted by \( \Phi_n(x) \), is given by the formula \( \Phi_n(x) = \prod_{d \mid n} (x^d - 1)^{\mu(n/d)} \), where \( \mu \) is the Möbius function. Cyclotomic polynomials play a crucial role in algebraic number theory as they help in constructing cyclotomic fields and understanding the structure of field extensions generated by roots of unity.
Dedekind's Criterion: Dedekind's Criterion provides a way to determine whether a given number field is a Dedekind domain by examining the factorization of ideals in its ring of integers. This criterion connects algebraic integers, number fields, and the behavior of prime ideals within those fields, highlighting the relationship between algebraic structures and their integral bases.
Degree of a field extension: The degree of a field extension is the dimension of the larger field as a vector space over the smaller field. It essentially quantifies how 'big' the extension is in relation to the base field and reveals significant information about the structure of both fields involved. This concept is crucial in understanding the behavior of roots of polynomials and the relationships between different fields, particularly when examining cyclotomic fields or analyzing Galois groups and their correspondence.
Factorization: Factorization is the process of breaking down an object, such as a number or polynomial, into a product of simpler objects called factors. This concept plays a pivotal role in various mathematical areas, revealing the underlying structure of numbers and providing insights into divisibility and properties of integers. Understanding factorization is essential for exploring unique representations of numbers, integral bases in number fields, and specialized number systems like Gaussian and Eisenstein integers.
Galois Group: A Galois group is a mathematical structure that captures the symmetries of the roots of a polynomial equation, formed by the automorphisms of a field extension that fix the base field. This concept helps us understand how different roots relate to one another and provides a powerful framework for analyzing the solvability of polynomials and the structure of number fields.
Kummer's Theorem: Kummer's Theorem provides a way to understand the behavior of primes in the context of cyclotomic fields, specifically relating to how these primes split in these fields. It highlights the relationship between the ramification of primes and their decomposition into distinct prime factors, which is crucial when studying the structure of cyclotomic fields and understanding the decomposition and inertia groups associated with primes in number theory.
Minimal Polynomial: The minimal polynomial of an algebraic element over a field is the monic polynomial of smallest degree that has the element as a root. This polynomial captures the essence of the element's algebraic properties and relates closely to the structure of number fields, field extensions, and their algebraic closures.
N-th roots of unity: The n-th roots of unity are the complex numbers that satisfy the equation $$z^n = 1$$, where $$n$$ is a positive integer. These roots are evenly spaced around the unit circle in the complex plane and can be expressed in exponential form as $$z_k = e^{2\\pi i k/n}$$ for integers $$k$$ from 0 to $$n-1$$. This concept is fundamental in the study of cyclotomic fields and cyclotomic polynomials, as these roots play a crucial role in their structure and properties.
Niels Henrik Abel: Niels Henrik Abel was a Norwegian mathematician known for his groundbreaking contributions to algebra, particularly in the area of group theory and elliptic functions. His work laid the foundation for many concepts in modern mathematics, including developments in Galois theory, which connects to the study of polynomial equations and their solutions through group symmetries.
Norm mapping: Norm mapping is a concept that describes the process of evaluating how elements in a number field relate to their 'norm', which is essentially a measure of size or magnitude. In the context of cyclotomic fields and cyclotomic polynomials, norm mapping helps in understanding the behavior of roots of unity and their corresponding polynomial equations, particularly when analyzing how these norms interact under field extensions.
Primitive Root of Unity: A primitive root of unity is a complex number that is a solution to the equation $$x^n = 1$$, and cannot be expressed as any lower power of another root of unity. These roots play a crucial role in various fields of mathematics, particularly in the construction of cyclotomic fields and the analysis of cyclotomic polynomials. They represent the vertices of a regular n-gon in the complex plane and are used to express other roots of unity through their powers.
Q(√d) for d = -1: The term q(√d) for d = -1 refers to the field extension created by adjoining the square root of -1, which is the imaginary unit 'i', to the rational numbers. This results in the complex number field, specifically denoted as q(i), and is crucial in understanding cyclotomic fields where roots of unity play a key role in defining their structure and properties.
Q(ζ_n): In algebraic number theory, q(ζ_n) refers to the field extension generated by a primitive n-th root of unity, denoted as ζ_n. This concept is crucial for understanding cyclotomic fields, which are generated by adjoining roots of unity to the rational numbers. Cyclotomic fields play a significant role in various areas, including number theory, algebra, and the study of Galois groups.
Ring of Integers: The ring of integers is the set of algebraic integers in a number field, which forms a ring under the usual operations of addition and multiplication. This concept is crucial as it provides a framework for studying the properties and behaviors of numbers in various algebraic contexts, particularly when dealing with number fields, discriminants, and integral bases.
Unit Group: The unit group of a ring is the set of elements that have multiplicative inverses within that ring. Understanding unit groups is crucial for exploring the structure of algebraic objects, particularly in relation to the behavior of integers and their generalizations, which often manifest in the study of rings and fields.