Finite field/Exponential function/Linearly independent/Exercise

Let ${}K$ be a finite field with ${}q$ elements.

1. Show that the polynomial functions

   $\varphi _{d}\colon K\longrightarrow K,x\longmapsto x^{d},$

   with ${}0\leq d<q$ are linearly independent.

2. Show that the exponential functions

   $\psi _{b}\colon K\longrightarrow K,x\longmapsto b^{x},$

   with ${}0\leq b<q$ are linearly independent.

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