Theoretical Backgrounds of Applied Discrete Mathematics

An algorithm for computing the Stickelberger ideal for multiquadratic number fields

E. A. Kirshanova, E. S. Malygina, S. A. Novoselov, D. O. Olefirenko

Immanuel Kant Baltic Federal University, Kaliningrad, Russia

Abstract: We present an algorithm for computing the Stickelberger ideal for multiquadratic fields $K=\mathbb{Q}(\sqrt{d_1}, \sqrt{d_2},\ldots,\sqrt{d_n})$, where the integers $d_i \equiv 1 \bmod 4$ for $i \in \{1, \ldots, n\} $ or $d_j \equiv 2 \bmod 8$ for one $j \in \{1, \ldots, n \}$; all $d_i$'s are pairwise co-prime and square-free. Our result is based on the paper of Kučera [J. Number Theory, no. 56, 1996]. The algorithm we present works in time $\mathcal{O}(\lg \Delta_K \cdot 2^n \cdot \mathrm{poly}(n) )$, where $\Delta_K$ is the discriminant of $K$. As an interesting application, we show a connection between Stickelberger ideal and the class number of a multiquadratic field.

Keywords: multiquadratic number field, Stickelberger element, Stickelberger ideal, class group of multiquadratic field.

UDC: 511.23

DOI: 10.17223/20710410/51/1

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