Finite field F_(p^k)

For a prime $p$ and an integer $k\ge1$ , $\mathbb F_{p^k}$ denotes a field with $p^k$ elements. It can be realized as the roots of $x^{p^k}-x$ in a chosen algebraic closure of $\mathbb F_p$ , or as $\mathbb F_p[x]/(f)$ for a chosen monic irreducible polynomial $f$ of degree $k$ .

Maps and field choices

The field is unique up to isomorphism, not up to a unique isomorphism. Its $\mathbb F_p$ -automorphism group is cyclic of order $k$ , generated by Frobenius $a\mapsto a^p$ . A polynomial presentation specifies coordinates; embeddings into another finite field are extra relationship data.

Degree and cardinality

The exponent $k$ is the vector-space dimension over $\mathbb F_p$ , whereas $p^k$ is the number of elements. This notation must not be confused with the residue ring $\mathbb Z/p^k\mathbb Z$ , which has zero divisors when the exponent exceeds one.

References

J. S. Milne, Fields and Galois Theory, Chapter 4, Propositions 4.19–4.23, pp. 53–54.