Field F_8 - Knowlpedia

The field $\mathbb F_8$ is the finite field presented by

\mathbb F_8=\mathbb F_2[x]/(x^3+x+1).

Writing $\alpha$ for the residue class of $x$ , every element has a unique representative of degree less than $3$ , and $\alpha^3=\alpha+1$ .

Why this is a field

The polynomial $x^3+x+1$ has no root in $\mathbb F_2$ . Since its degree is three, it is irreducible; the quotient is therefore a field. Counting coefficients gives exactly 8 elements.

Field maps versus linear maps

The field automorphism group has order $3$ , generated by $a\mapsto a^2$ . The additive space has dimension $3$ over $\mathbb F_2$ , so its linear automorphism group is $\mathrm{GL}_3(\mathbb F_2)$ . A general invertible linear map need not preserve products or the unit.

Recorded linear maps

Its linear endomorphism ring over $\mathbb F_2$ is $M_3(\mathbb F_2)$ , with 512 maps. Its invertible linear maps form $\mathrm{GL}_3(\mathbb F_2)$ , with 168 elements. The catalogue keeps this scalar category separate from field morphisms, for which only the 3 Frobenius powers occur.

References

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