  This work explores how degrees of freedom (DoF) results from wireless
networks can be translated into capacity results for their finite field
counterparts that arise in network coding applications. The main insight is
that scalar (SISO) finite field channels over $\mathbb{F}_{p^n}$ are analogous
to n x n vector (MIMO) channels in the wireless setting, but with an important
distinction -- there is additional structure due to finite field arithmetic
which enforces commutativity of matrix multiplication and limits the channel
diversity to n, making these channels similar to diagonal channels in the
wireless setting. Within the limits imposed by the channel structure, the DoF
optimal precoding solutions for wireless networks can be translated into
capacity optimal solutions for their finite field counterparts. This is shown
through the study of the 2-user X channel and the 3-user interference channel.
Besides bringing the insights from wireless networks into network coding
applications, the study of finite field networks over $\mathbb{F}_{p^n}$ also
touches upon important open problems in wireless networks (finite SNR, finite
diversity scenarios) through interesting parallels between p and SNR, and n and
diversity.
