Cyclotomic Field

Definition

A cyclotomic field is a special type of number field obtained by adjoining a primitive root of unity to the rational numbers. These fields are generated by the roots of the polynomial equation $x^n - 1 = 0$, where $n$ is a positive integer, and they play a key role in number theory, particularly in understanding the properties of integers and their divisibility.

5 Must Know Facts For Your Next Test

1. Cyclotomic fields are generated by adding a primitive $n$-th root of unity, denoted as $\zeta_n = e^{2\pi i/n}$, to the rationals, creating $\mathbb{Q}(\zeta_n)$.
2. The degree of a cyclotomic field over the rationals is given by Euler's totient function $\varphi(n)$, which counts the number of integers up to $n$ that are coprime to $n$.
3. The Galois group of a cyclotomic field is isomorphic to $(\mathbb{Z}/n\mathbb{Z})^*$, the multiplicative group of integers modulo $n$, reflecting the structure of its roots.
4. Cyclotomic fields are abelian extensions of the rationals, which means their Galois groups are commutative, leading to important implications in class field theory.
5. Many results in number theory, such as Fermat's Last Theorem and properties related to modular forms, can be connected to cyclotomic fields and their extensions.

Review Questions

• How do cyclotomic fields relate to primitive roots of unity and what implications does this relationship have on their structure?
  • Cyclotomic fields are constructed by adjoining primitive roots of unity, specifically $\\zeta_n$, which leads to the formation of fields such as $\\mathbb{Q}(\\zeta_n)$. This relationship highlights how these fields encapsulate the symmetry of roots and their interrelations through the Galois group, which provides insights into the nature of divisibility and factorization within these number fields. Understanding this structure is fundamental for exploring deeper concepts in number theory.
• Discuss how Euler's totient function influences the degree of cyclotomic fields over the rationals and its significance in algebraic number theory.
  • Euler's totient function $\\varphi(n)$ plays a crucial role in determining the degree of a cyclotomic field over the rationals. The degree reflects how many distinct roots can be generated from $\\zeta_n$, and understanding this degree is vital for classifying extensions and examining their properties within algebraic number theory. The relationship illustrates how cyclotomic fields serve as building blocks for larger structures, affecting everything from divisibility to Galois theory.
• Analyze the role cyclotomic fields play in modern number theory, especially regarding abelian extensions and their connections to classical results.
  • Cyclotomic fields are essential in modern number theory as they exemplify abelian extensions, which link closely with Galois theory and class field theory. The Galois group's commutative nature offers insights into various problems, including those surrounding Fermat's Last Theorem and properties of modular forms. Their structures enable mathematicians to explore deep relationships between different areas in number theory, influencing both theoretical aspects and practical applications involving cryptography and primality testing.

Related terms

Primitive Root of Unity: A complex number that is a solution to the equation $x^n = 1$ and is not a solution to $x^k = 1$ for any positive integer $k < n$.

Galois Group:

An algebraic structure that describes the symmetries of the roots of a polynomial, particularly in relation to field extensions.

Algebraic Integers:

Elements of a number field that are roots of monic polynomials with integer coefficients, extending the concept of integers to include more numbers.