Exercise 1, Appendix 0

Lie Groups and Lie Algebras, II, 207

Let $\Phi_n$ be the cyclotomic polynomial of index $n$ (Algebra, Chapter V, § 11, no. 2). Using the formula

X^n - 1 = \prod_{d|n} \Phi_d(X),

show that

\Phi_n(X) = \prod_{d|n} (X^{n/d} - 1)^{\mu(d)}.

(Apply the Möbius inversion formula to the multiplicative group of the field

\mathbf{Q}(X)

Solution

No solution has been written yet.