Field F_27 - Knowlpedia

The field $\mathbb F_{27}$ is the finite field

\mathbb F_{27}=\mathbb F_3[x]/(x^3-x-1).

With $\alpha$ the residue class of $x$ , its elements are uniquely $a+b\alpha+c\alpha^2$ , with $a,b,c\in\mathbb F_3$ , and multiplication uses $\alpha^3=\alpha+1$ .

Irreducibility

Each of the three substitutions $x=0,1,2$ gives the nonzero residue $2$ . A reducible cubic over a field has a linear factor, hence a root, so this polynomial is irreducible. The quotient is a field and the three independent coefficients give 27 elements.

Automorphisms and linear maps

Frobenius $a\mapsto a^3$ generates the field automorphism group of order three. In contrast, the invertible $\mathbb F_3$ -linear maps form $\mathrm{GL}_3(\mathbb F_3)$ . A linear map need not fix the unit or preserve the defining multiplication.

References

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