What is the Galois group of a polynomial over a finite field?

Are group theory, field theory, Galois theory or category theory useful in theoretical economics?

• For example, Galois connections have made an appearance in a recent paper on implementation by Noldeke and Samuelson: http://www.wiso.tu-dortmund.de/wiso/mik/Medienpool/theoretischer_ausschuss/downloads/2013/TA13Noeldeke.pdf.

• Answer:

  In the context of that paper, a "Galois connection" simply refers to an order-reversing correspondence of posets.  So it's quite unlikely that knowing anything nontrivial about Galois theory would be helpful in understanding that paper. In a broader sense, it's hard to say what kind of math is relevant to what kind of real-world problem.  Prior to the 20th century most people wouldn't have thought that group theory was particularly relevant to fundamental physics.  Current mainstream economic models aren't closely tied to algebra, but of course it's well known that current models have very serious shortcomings and perhaps fixing these will require a dramatically different class of model that would be amenable to analysis by other kinds of math.