paljastupolynomijaksoja

Paljastupolynomijaksoja, known in English as cyclotomic polynomials, are a fundamental concept in abstract algebra and number theory. They are irreducible polynomials whose roots are the primitive $n$-th roots of unity for a given positive integer $n$.

The $n$-th cyclotomic polynomial, denoted by $\Phi_n(x)$, is defined as the polynomial whose roots are precisely the complex numbers $\zeta$ such that $\zeta^n = 1$ and $\zeta^k \neq 1$ for any $1 \le k < n$. These roots are called the primitive $n$-th roots of unity. The degree of $\Phi_n(x)$ is equal to $\phi(n)$, where $\phi$ is Euler's totient function, which counts the number of positive integers less than or equal to $n$ that are relatively prime to $n$.

A key property of cyclotomic polynomials is their relationship with the polynomial $x^n - 1$. Specifically, $x^n - 1$ can be factored into a product of cyclotomic polynomials: $x^n - 1 = \prod_{d|n} \Phi_d(x)$, where the product is taken over all positive divisors $d$ of $n$. This factorization allows for the recursive determination of $\Phi_n(x)$ using the formula $\Phi_n(x) = \frac{x^n - 1}{\prod_{d|n, d<n} \Phi_d(x)}$.

Cyclotomic polynomials are always irreducible over the field of rational numbers $\mathbb{Q}$. This irreducibility is crucial in algebraic number theory, particularly in the study of cyclotomic fields, which are fields obtained by adjoining the $n$-th roots of unity to the field of rational numbers. These polynomials have applications in various areas, including the construction of regular polygons, coding theory, and cryptography.

---