I am trying to understand the concept of Galois fields from a beginner level, I read that Galois fields are of the form $GF(p^{n})$.
- What difference does it make conceptually to the properties of a field?
- Also why Galois extension field elements considered as polynomials?
$GF(p^n)$ is the set of polynomials of degree less than $n$, whose $n$ coefficients are all in $GF(p)$. Multiplication of polynomials is taken modulo $f(x)$, where $f(x)$ is a polynomial of degree $n$ without any factors.
Note $p=0$ in both $GF(p)$ and $GF(p^n)$.