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

• Cyclotomic fields form by adjoining roots of unity to rational numbers Q
• Nth cyclotomic field 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 Galois group over Q abelian
• Ring of integers 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 $\pm n^{n\phi(n)}/\prod_{p|n} p^{p\phi(n)/(p-1)}$ (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 cyclotomic polynomial Φn(x) serves as minimal polynomial of primitive nth root of unity over Q
• Recursive construction using formula $x^n - 1 = \prod_{d|n} \Phi_d(x)$ (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: $\Phi_p(x) = x^{p-1} + x^{p-2} + ... + x + 1$

Properties and Examples

• Coefficients of cyclotomic polynomials always integers (not obvious from definition)
• First few cyclotomic polynomials:
  • $\Phi_1(x) = x - 1$
  • $\Phi_2(x) = x + 1$
  • $\Phi_3(x) = x^2 + x + 1$
  • $\Phi_4(x) = x^2 + 1$
• Möbius inversion formula used to express Φn(x) explicitly
• Cyclotomic polynomials satisfy various identities ($\Phi_n(-x) = \pm \Phi_n(x)$ 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 factorization 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