Consider as an element of and use bars to denote the natural projection .

- Fine a polynomial of degree at most 3 which is congruent to modulo .
- Prove that and are zero divisors in .

- Evidently, .
- Note that . By §7.4 #14, and are zero divisors in .

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## Comments

can you tell me how you’re doing these? there’s nothing like this in the book

There’s a couple of ways to think about this problem.

Remember that two polynomials and are equivalent mod if divides . Using the Division Algorithm in the Euclidean domain , we can find and such that , and the degree of is less than that of ; this is the (in this case unique) representative of the coset with this property.

Sometimes, it is easier to reduce powers of directly. For example, in this ring we have ; so . In this fashion, we can eliminate powers of larger than the degree of .

I think there is a typo in part 2. The last term of the factorization of x^4-16 should read (x^2+4).

Fixed. Thanks!