Cyclotomic field

Definition

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.

5 Must Know Facts For Your Next Test

1. The cyclotomic field $\mathbb{Q}(\zeta_n)$ is generated by adding $\zeta_n$ to $\mathbb{Q}$, which results in an extension of degree $\varphi(n)$, where $\varphi$ is Euler's totient function.
2. The Galois group of a cyclotomic field is isomorphic to the multiplicative group of integers modulo $n$, reflecting the symmetry among the roots of unity.
3. Cyclotomic fields are abelian extensions, meaning their Galois groups are abelian, which allows for powerful results like the Kronecker-Weber theorem, stating that every abelian extension can be obtained from cyclotomic fields.
4. The ring of integers in a cyclotomic field is often generated by elements related to powers of $\zeta_n$, providing useful integral bases for computations.
5. Cyclotomic fields have applications in various areas such as class field theory, algebraic geometry, and cryptography due to their rich structure and properties.

Review Questions

• How do cyclotomic fields illustrate the applications of Galois theory in understanding extensions of the rational numbers?
  • Cyclotomic fields serve as prime examples in Galois theory because their Galois groups can be explicitly described and analyzed. By examining these fields, we can gain insights into how field extensions behave under automorphisms. The structure of these Galois groups reveals symmetries related to roots of unity, showcasing how Galois theory helps us understand not just cyclotomic fields but also more complex number fields.
• Discuss the significance of integral bases within cyclotomic fields and how they contribute to our understanding of their properties.
  • Integral bases in cyclotomic fields help define the ring of integers within these extensions. By identifying a suitable integral basis, we can simplify calculations involving ideal classes and factorization. This clarity in structure enables mathematicians to apply results from algebraic number theory effectively, linking cyclotomic fields to broader concepts such as class numbers and units in rings.
• Evaluate how cyclotomic polynomials relate to cyclotomic fields and their implications on number theory, particularly with respect to root structures and factorization.
  • Cyclotomic polynomials are foundational to understanding cyclotomic fields as they define the minimal polynomials for primitive $n$th roots of unity. The factorization properties of these polynomials over different fields indicate how roots behave under various conditions. This relationship between polynomials and their corresponding fields informs deeper aspects of number theory, including results on divisibility, primes in number fields, and even connections to modern cryptographic methods.

Related terms

Galois group: The Galois group of a field extension is the group of field automorphisms that fix the base field and describes the symmetries in the roots of polynomials.

Integral basis: An integral basis for a number field is a set of elements that generates the ring of integers of that field as an additive group and consists of integral elements over the base field.

Cyclotomic polynomial: A cyclotomic polynomial is a polynomial whose roots are the primitive $n$th roots of unity, expressed as $\Phi_n(x)$, and plays a key role in constructing cyclotomic fields.