  Higher-order Fourier analysis, developed over prime fields, has been recently
used in different areas of computer science, including list decoding,
algorithmic decomposition and testing. We extend the tools of higher-order
Fourier analysis to analyze functions over general fields. Using these new
tools, we revisit the results in the above areas.
  * For any fixed finite field $\mathbb{K}$, we show that the list decoding
radius of the generalized Reed Muller code over $\mathbb{K}$ equals the minimum
distance of the code. Previously, this had been proved over prime fields [BL14]
and for the case when $|\mathbb{K}|-1$ divides the order of the code [GKZ08].
  * For any fixed finite field $\mathbb{K}$, we give a polynomial time
algorithm to decide whether a given polynomial $P: \mathbb{K}^n \to \mathbb{K}$
can be decomposed as a particular composition of lesser degree polynomials.
This had been previously established over prime fields [Bha14, BHT15].
  * For any fixed finite field $\mathbb{K}$, we prove that all locally
characterized affine-invariant properties of functions $f: \mathbb{K}^n \to
\mathbb{K}$ are testable with one-sided error. The same result was known when
$\mathbb{K}$ is prime [BFHHL13] and when the property is linear [KS08].
Moreover, we show that for any fixed finite field $\mathbb{F}$, an
affine-invariant property of functions $f: \mathbb{K}^n \to \mathbb{F}$, where
$\mathbb{K}$ is a growing field extension over $\mathbb{F}$, is testable if it
is locally characterized by constraints of bounded weight.
