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This paper presents an efficient algorithm for computing 11th-power residue symbols in the cyclo-tomic field ℚ (ζ 11 ) , $ \mathbb{Q}\left({{\zeta }_{11}} \right), $ where 11 is a primitive 11th root of unity. It extends an earlier algorithm due to Caranay and Scheidler (Int. J. Number Theory, 2010) for the 7th-power residue symbol. The new algorithm finds applications in the implementation of certain cryptographic schemes. [ABSTRACT FROM AUTHOR]

We present an infinite series formula based on the Karoubi-Hamida integral, for the universal Borel class evaluated on H2 n+1(GL(ℂ)). For a cyclotomic field F we define a canonical set of elements in K3( F) and present a novel approach (based on a free differential calculus) to constructing them. Indeed, we are able to explicitly construct their images in H3(GL(ℂ)) under the Hurewicz map. Applying our formula to these images yields a value V1( F), which coincides with the Borel regulator R1( F) when our set is a basis of K3( F) modulo torsion. For F = ℚ( e2π i/3) a computation of V1( F) has been made based on our techniques. [ABSTRACT FROM AUTHOR]

We show that for any integers n ≥≥ 2 and 0 ≤≤ k ≤≤ n, one may find k distinct nth roots of unity whose sum is 0 if and only if both k and n -- k are expressible as linear combinations of prime factors of n with nonnegative coefficients. [ABSTRACT FROM AUTHOR]