Visualising finite fields

Solution 1:

Here's an interesting take on the problem of visualizing finite fields: Thomas Alden Gassert, a graduate student, studies finite fields generated by the solutions of iterated polynomials, yielding a visualization of finite fields via cycles. See the pictures on his website and especially in his papers: https://www.math.umass.edu/~gassert/

Solution 2:

While not giving visualizations of finite fields you might find the book: A Geometrical Picture Book by Burkard Polster, Springer, NY, 1998 of interest. It has ways of visualizing various combinatorial objects such as finite affine planes and finite projective planes (some of which can be co-ordinatized using finite fields), as well as means for drawing other finite combinatorial objects (Steiner triple systems, generalize quadrangles, etc.).