This set is due **April 10** at the beginning of lecture. Details of the homework policy can be found on the syllabus and here.

**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 )

[…] Homework 7, due April 10, at the beginning of lecture. […]

Hi, I finished problem 1, and looking back I just want to make sure the assumptions I make are valid. For all three parts of 1, am I allowed to assume that the unit i is in I? I assume I am, but I don’t want to lose points for something so silly. Thanks

I am a bit confused concerning problem 4. Is it even possible to build such a field? I know I can construct a field of 27 elements, but to construct a field of 9 elements with a characteristic of 3, wouldn’t the polynomial have to be quadratic? Thanks again!

Hi Tommy,

I’m not sure I understand your question on problem 1. Email me, and tell me in some detail what you are doing, just to be sure.

About problem 4, you are correct: If we are to build a field of 9 elements, then the polynomial in we need to consider must be quadratic and irreducible.