Minkowski's Linear Forms Theorem is often stated about linear forms with real coefficients. However, in Narkiewicz's *Elementary and Analytic Theory of Algebraic Numbers*, the following generalization of Minkowski's Linear Forms Theorem is stated (the theorem is actually stated for a general lattice in real space, but I'm only interested in the case where the lattice is $\mathbb{Z}^N$):

Let $U = \{ L_j(x_1, \ldots, x_N) = \sum_{i=1}^N a_{ij} x_i \mid 1 \leq j \leq N \}$ be a system of $N$ linear forms with complex coefficients satisfying the following condition: if $L_j \in U$, then the conjugate form, $\overline{L_j} \in U$. Let $M = [a_{jk}]_{1\leq j,k \leq N}$. Let $\epsilon_1, \ldots, \epsilon_N \in \mathbb{R}^+$ such that $\prod_{i=1}^N \epsilon_i \geq | \det M |$ and $L_i = \overline{L_j}$ implies $\epsilon_i = \epsilon_j$. Then there exists a nonzero point $(x_1, \ldots, x_n) \in \mathbb{Z}^N$ such that $|L_j(x_1,\ldots,x_N)| \leq \epsilon_j$ for each $1 \leq j \leq N$ with strict inequality holding for all but one $j$.

However, instead of providing a proof, Narkiewicz refers to two books by Cassels, both of which only prove the case where the coefficients of the forms are real. The proof of the real version---which uses Minkowski's Convex Body Theorem on $M^{-1} ([-1,1]^N)$---does not easily generalize to the complex version. Does anyone have an idea of how the proof of the generalization might go? I've tried translating the problem to real-space (using the fact that $N$-dimensional complex space is isomorphic to $2N$-dimensional real space), but the set that I want to apply Minkowski's Convex Body Theorem to does not have large enough volume.

EDIT: A proof may appear in the 39th "chapter" in Minkowski's *Geometrie der Zahlen* (on page 113). However, I cannot read German. Does anyone know of an English translation of the work?

Lectures on the theory of algebraic numbers(Theorem 95, pp. 104--105 in the English translation). I can currently access this on Google Books: books.google.com/… $\endgroup$ – so-called friend Don Mar 21 '17 at 0:36