Q(ζ_n)

Definition

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.

5 Must Know Facts For Your Next Test

1. The cyclotomic field Q(ζ_n) is obtained by adjoining ζ_n to the field of rational numbers Q.
2. The degree of the extension [Q(ζ_n):Q] is equal to φ(n), where φ is Euler's totient function, counting the integers up to n that are coprime to n.
3. The Galois group of the cyclotomic field Q(ζ_n) is isomorphic to (Z/nZ)*, the multiplicative group of integers modulo n.
4. Cyclotomic fields are normal extensions because they include all roots of their defining polynomial, which is a cyclotomic polynomial.
5. Every abelian extension of Q can be realized as a subfield of some cyclotomic field.

Review Questions

• How does q(ζ_n) illustrate the relationship between roots of unity and field extensions?
  • q(ζ_n) exemplifies this relationship by showcasing how primitive n-th roots of unity can generate a new field, known as a cyclotomic field. When ζ_n is adjoined to Q, it creates a larger field that contains all powers of ζ_n and demonstrates how these roots help form the underlying structure of number fields. This illustrates how algebraic entities can produce extensions that enrich our understanding of number theory.
• Discuss the significance of the Galois group associated with q(ζ_n) and its implications for understanding its structure.
  • The Galois group associated with q(ζ_n) is pivotal as it reveals symmetries and relationships between different roots within the cyclotomic field. Since this group is isomorphic to (Z/nZ)*, it highlights how each element corresponds to a distinct automorphism that preserves the structure of the field while mapping roots of unity to each other. Understanding this group allows for deeper insights into properties such as solvability and classification of polynomial equations related to q(ζ_n).
• Evaluate how q(ζ_n) connects with broader concepts in algebraic number theory and its applications.
  • q(ζ_n) serves as a cornerstone in algebraic number theory, linking cyclotomic fields to various important concepts such as class field theory and L-functions. This connection allows mathematicians to explore questions related to primes in arithmetic progressions and other number-theoretic properties. Furthermore, since every abelian extension can be realized through these fields, q(ζ_n) becomes essential in applying theoretical results to real-world problems like cryptography and coding theory.

Related terms

Cyclotomic Polynomial: A polynomial whose roots are the primitive n-th roots of unity, given by the formula \(\Phi_n(x) = \prod_{d|n}(x^d - 1)^{\mu(n/d)}\).

Galois Group: The group of field automorphisms of a field extension that fix the base field, reflecting the symmetry of the roots of polynomials in that extension.

Field Extension: A larger field containing a smaller field as a subfield, which allows for the study of solutions to polynomial equations that may not exist in the smaller field.