  We propose a generalization of the graphical ZH calculus to qudits of
prime-power dimensions $q = p^t$, implementing field arithmetic in arbitrary
finite fields. This is an extension of a previous result by Roy which
implemented arithmetic of prime-sized fields; and an alternative to a result by
de Beaudrap which extended the ZH to implement cyclic ring arithmetic in
$\mathbb Z / q\mathbb Z$ rather than field arithmetic in $\mathbb F_q$. We show
this generalized ZH calculus to be universal over matrices $\mathbb C^{q^n} \to
\mathbb C^{q^m}$ with entries in the ring $\mathbb Z[\omega]$ where $\omega$ is
a $p$th root of unity. As an illustration of the necessity of such an extension
of ZH for field rather than cyclic ring arithmetic, we offer a graphical
description and proof for a quantum algorithm for polynomial interpolation.
This algorithm relies on the invertibility of multiplication, and therefore can
only be described in a graphical language that implements field, rather than
ring, multiplication.
