Cyclotomic polynomial

Definition

A cyclotomic polynomial is a special type of polynomial defined as the product of linear factors corresponding to the primitive roots of unity. Specifically, the nth cyclotomic polynomial, denoted by \( \Phi_n(x) \), is given by the formula \( \Phi_n(x) = \prod_{d \mid n} (x^d - 1)^{\mu(n/d)} \), where \( \mu \) is the Möbius function. Cyclotomic polynomials play a crucial role in algebraic number theory as they help in constructing cyclotomic fields and understanding the structure of field extensions generated by roots of unity.

5 Must Know Facts For Your Next Test

1. The degree of the nth cyclotomic polynomial is given by Euler's totient function, \( \varphi(n) \), which counts the integers up to n that are coprime to n.
2. Cyclotomic polynomials are irreducible over the integers, meaning they cannot be factored into lower-degree polynomials with integer coefficients.
3. The coefficients of cyclotomic polynomials are integers and exhibit a symmetric pattern based on their roots.
4. The cyclotomic polynomial \( \Phi_n(x) \) is related to the roots of unity, specifically providing the minimal polynomial for these roots over the rationals.
5. Cyclotomic fields have unique properties, such as being abelian extensions of the rational numbers, which have implications in Galois theory.

Review Questions

• How do cyclotomic polynomials relate to primitive roots of unity and what is their significance in algebraic number theory?
  • Cyclotomic polynomials are directly constructed from primitive roots of unity, as they encapsulate the roots corresponding to these units. They are significant because they provide a way to understand how these roots generate field extensions known as cyclotomic fields. These fields facilitate deeper exploration into the arithmetic properties and structures within number theory.
• Analyze how the irreducibility of cyclotomic polynomials over the integers impacts their use in constructing number fields.
  • The irreducibility of cyclotomic polynomials over the integers means that they serve as minimal polynomials for primitive roots of unity. This property ensures that when we construct cyclotomic fields using these polynomials, we create extensions that maintain essential arithmetic properties. As such, they allow mathematicians to study various algebraic structures without losing information about these roots.
• Evaluate the implications of Euler's totient function on the degrees of cyclotomic polynomials and their relationship with field extensions.
  • Euler's totient function determines the degree of each cyclotomic polynomial, which directly impacts the dimension of corresponding cyclotomic fields as vector spaces over the rationals. By analyzing how \( \, \varphi(n) \) varies with n, one can understand how complex roots interact within these fields and explore properties like Galois groups. The behavior of these degrees sheds light on both algebraic structure and symmetry present in number theory.

Related terms

primitive root of unity: A complex number that is a solution to the equation \( z^n = 1 \) and is not a solution to \( z^k = 1 \) for any proper divisor \( k < n \).

cyclotomic field: A number field obtained by adjoining a primitive nth root of unity to the rational numbers, denoted as \( \mathbb{Q}(\zeta_n) \), where \( \zeta_n = e^{2\pi i/n} \).

Möbius function: A function that assigns values based on the factorization of integers; it is defined as \( \mu(n) = 1 \) if n is a product of an even number of distinct primes, \( -1 \) if odd, and 0 if n has a squared prime factor.