## Exhibit some finite fields

Let and . Construct finite fields of order 4, 8, 9, and 27 elements. For the fields with 4 and 9 elements, give the multiplication tables and show that the nonzero elements form a cyclic group.

Note that to construct fields of the given sizes, it suffices to show that and are each irreducible over the fields and . (More precisely, that these polynomials are irreducible when we inject the coefficients into the respective fields.)

It is easy to see that neither nor has a root in either or in , so these are indeed irreducible. Then , , , and are fields of order 4, 8, 9, and 27 by Proposition 11 in D&F, and specifically are isomorphic as fields to the extension of or by adjoining a root of or .

The multiplication table of is as follows.

Evidently, is generated by .

The multiplication table of is as follows.

Evidently, is generated by .

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