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Let p > 3 be a prime number, ζ be a primitive p-th root of unity. Suppose that the Kummer-Vandiver conjecture holds for p , i.e., that p does not divide the class number of Q (ζ + ζ - 1) . Let λ and ν be the Iwasawa invariants of Q (ζ) and put λ = : ∑ i ∈ I λ i and ν = : ∑ i ∈ I ν i in the usual way, where I is the set of odd integers in { 3 , ... , p - 2 } . We prove for each i ∈ I that λ i ≤ 1 , so that the p-primary part of class group of each Q (exp (2 π - 1 / p n)) is determined by the set { (i , ν i) : i ∈ I } . [ABSTRACT FROM AUTHOR]

Complex periodical sequences with lower autocorrelation values are used in CDMA communication systems and cryptography. In this paper we present new nonexistence results on perfect p-ary sequences and almost p-ary sequences and related difference sets by using some knowledge on cyclotomic fields and their subfields. [ABSTRACT FROM AUTHOR]

Recently the authors [] showed that the algebraic integers of the form $${-m+\zeta_k}$$ are bases of a canonical number system of $${\mathbb {Z}[{\zeta_k]}}$$ provided $${m \geqq \phi(k)+1}$$ , where $${\zeta_k}$$ denotes a k-th primitive root of unity and $${\phi}$$ is Euler's totient function. In this paper we are interested in the questions whether two bases $${-m+\zeta_k}$$ and $${-n+\zeta_k}$$ are multiplicatively independent. We show the multiplicative independence in case that 0 < | m− n| < 10 and | m|, | n| > 1. [ABSTRACT FROM AUTHOR]