If f(x) = let f(x⁵) be the polynomial

Establish the following i=O i=O properties of the cyclotomic polynomials g_n(x) over Q.

(a) If p is prime and k ≥ 1, then g_pk(x) = g_p(x^Pk-1).

(b) If n = p₁^r1• ·P_k^rk (p_i distinct primes; r_i > O), then 

(c) If n is odd, then g_2n(X) = g_n(-x).

(d) If p is a prime and płn, then g_pn(x) = g_n(x^P)/g_n(x) 

(e) gn(x) = where µ is the Moebius function of Exercise 2(e).

(f) g_n(1) = p if n = P^k (k > 0), 0 if n = I, and 1 otherwise.

Data from exercise 2(e) 

Transcribed Image Text:

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