1. Let be a commutative ring with identity. Let be an ideal of and let be a unit of Show that iff Conclude that the only ideals of a field are and Also conclude that if is the ideal of generated by a constant nonzero polynomial.
2. Suppose is a nonconstant, not irreducible polynomial, and let be the ideal generated by Show that has zero divisors.
3. Find an irreducible polynomial in of degree 3, and explicitly show that coincides with (i.e., is isomorphic to) the field of 8 elements built in a previous homework set.
4. Either build a field of 9 elements using the kinds of arguments in the previous problems; or determine a subfield of isomorphic to (You may assume that is irreducible in )