How to find a primitive polynomial for the construction of a Galois field $\mathbb{F}_{p^m}$?

Question

Say I need to construct the finite field $\mathbb{F}_{5^3}$ or $\mathbb{F}_{7^2}$, where do I get a primitive polynomial? Sorry I'm new to learning this and some of what I'm typing may not be making sense.

Answer 1

It's hard to say the best way to do this. There is some theory to help construct primitive polynomials (such as finding irreducible factors of certain cyclotomic polynomials, etc.), but I would say that it involves a decent knowledge about finite fields.

Usually if you are doing this as part of a homework assignment, you just check that a polynomial is primitive (this is not too hard for small extension fields). There are also tables to look up examples. (really, even in common practice among experts you would look these up in a table)