We introduce two new packages, Nemo and Hecke, written in the Julia programming language for computer algebra and number theory. We demonstrate that high performance generic algorithms can be implemented in Julia, without the need to resort to a low-level C implementation. For specialised algorithms, we use Julia's efficient native C interface to wrap existing C/C++ libraries such as Flint, Arb, Antic and Singular. We give examples of how to use Hecke and Nemo and discuss some algorithms that we have implemented to provide high performance basic arithmetic.

We present a variation of the modular algorithm for computing the Hermite Normal Form of an OK-module presented by Cohen [4], where OK is the ring of integers of a number field K. An approach presented in [4] based on reductions modulo ideals was conjectured to run in polynomial time by Cohen, but so far, no such proof was available in the literature. In this paper, we present a modification of the approach of [4] to prevent the coefficient swell and we rigorously assess its complexity with respect to the size of the input and the invariants of the field K.