| INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 7(2) (2007), #A25 IRREDUCIBLE RADICAL EXTENSIONS AND EULER-FUNCTION CHAINS (2008) | |||||||||||||
Abstract | |||||||||||||
| 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 k on a set of asymptotic density 1. 1. | |||||||||||||
Details der Publikation | |||||||||||||
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