kantapolynomi

Kantapolynomi is a term used in some contexts, particularly in Finnish mathematics, to refer to a polynomial whose coefficients are related to the roots of unity. Specifically, a kantapolynomi of a primitive $n$-th root of unity $\zeta$ is the minimal polynomial of $\zeta$ over the field of rational numbers. This minimal polynomial is also known as the $n$-th cyclotomic polynomial, denoted by $\Phi_n(x)$.

The cyclotomic polynomials are irreducible over the rational numbers and have integer coefficients. The degree of $\Phi_n(x)$ is $\phi(n)$, where $\phi$ is Euler's totient function. The product of all cyclotomic polynomials $\Phi_d(x)$ for all divisors $d$ of $n$ equals $x^n - 1$. That is, $x^n - 1 = \prod_{d|n} \Phi_d(x)$.

For example, the primitive cube root of unity is $\omega = e^{2\pi i / 3}$. Its minimal polynomial over the rationals is $x^2 + x + 1$, which is $\Phi_3(x)$. Similarly, the primitive fourth root of unity is $i = e^{2\pi i / 4}$. Its minimal polynomial is $x^2 + 1$, which is $\Phi_4(x)$. The concept of kantapolynomi is essentially a synonym for the cyclotomic polynomial in certain mathematical traditions. These polynomials play a significant role in number theory and abstract algebra, particularly in the study of field extensions and Galois theory.