Irreducible Radical Extensions and Euler-function Chains
نویسندگان
چکیده
We discuss the smallest algebraic number field which contains the nth roots of unity and which may be reached from the rational field Q by a sequence of irreducible, radical, Galois extensions. The degree D(n) of this field over Q is φ(m), where m is the smallest multiple of n divisible by each prime factor of φ(m). The prime factors of m/n are precisely the primes not dividing n but which do divide some number in the “Euler chain” φ(n),φ(φ(n)), . . . . For each fixed k, we show that D(n) > n on a set of asymptotic density 1. –For Ron Graham on his 70th birthday
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