Cyclotomic Fields

Definition

Cyclotomic fields are number fields generated by adjoining a primitive root of unity, typically denoted as $$\ ext{zeta}_n = e^{2\pi i / n}$$, where $$n$$ is a positive integer. These fields are significant in number theory, particularly in studying the properties of algebraic integers, Galois groups, and class numbers. Cyclotomic fields reveal deep connections between algebra and geometry through their roots of unity, which also impact discriminants and their calculations, field extensions, and ramification behavior.

5 Must Know Facts For Your Next Test

1. Cyclotomic fields are denoted as $$\mathbb{Q}(\zeta_n)$$ where $$\zeta_n$$ is a primitive nth root of unity.
2. The degree of the cyclotomic field extension $$\mathbb{Q}(\zeta_n)$$ over $$\mathbb{Q}$$ is given by Euler's totient function, $$\varphi(n)$$.
3. The discriminant of a cyclotomic field can be computed using the formula $$(-1)^{\varphi(n)/2} n^{\varphi(n)^{2}}$$.
4. Cyclotomic fields have unique factorization properties and their class numbers can be calculated using various techniques, including class number formulas.
5. Ramification in cyclotomic fields occurs at primes that divide $$n$$, leading to interesting behaviors in inertia groups and local field structures.

Review Questions

• How do cyclotomic fields relate to Galois theory, particularly regarding their Galois groups?
  • Cyclotomic fields are Galois extensions of the rational numbers, with their Galois group being isomorphic to the multiplicative group of integers modulo $$n$$. This connection illustrates how the symmetries in the roots of unity correspond to group elements representing automorphisms. The structure of the Galois group allows us to analyze the relationships between various cyclotomic fields and their subfields.
• Discuss how discriminants play a role in understanding cyclotomic fields and their extensions.
  • The discriminant of a cyclotomic field provides insight into the arithmetic properties of the field. Specifically, it helps determine whether the field is ramified or unramified at certain primes. The discriminant's value influences the field's ideal class group and informs calculations related to its class number. Thus, understanding discriminants is crucial for studying cyclotomic fields' algebraic structures.
• Evaluate the implications of cyclotomic fields on class numbers and their calculations within algebraic number theory.
  • Cyclotomic fields have significant implications for class number computations, often yielding results that are crucial for broader theories in algebraic number theory. The structure of cyclotomic fields often simplifies class number formulas due to their unique factorization properties and their well-defined discriminants. Evaluating these class numbers leads to further insights into the distribution of prime ideals in these fields, enhancing our understanding of algebraic integers' behavior in number theory.

Related terms

Primitive Root of Unity: A complex number that is a solution to the equation $$x^n = 1$$ and can be expressed as $$e^{2\pi i k/n}$$ for some integer $$k$$.

Galois Group: A group that describes the symmetries in the roots of polynomial equations, especially useful in understanding field extensions.

Discriminant: A quantity that provides important information about the roots of a polynomial, helping to determine the nature of the field extensions.