Hi

has no root in so it can't be written as a product of irreducible polynomials of degree 1 and 3. The other and last possibility for it to be reducible is to be the product of two irreducible polynomials of degree 2. The only irreducible polynomial of degree 2 in is (exercice), and we have

Therefore is reducible, and:

( )(wrong) which is not a field, since it's not an integral domain.

With the same idea of proof, you can show that is irreducible in and then, since this ring is a principal ideal domain, is a field.

and are fields, and

By unicity up to isomorphism for finite fields, and

In conclusion, ( )(wrong) while