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 .
Consider as an element of and use bars to denote the natural projection .
On these pages you will find a slowly growing (and poorly organized) list of proofs and examples in abstract algebra.
No doubt these pages are riddled with typos and errors in logic, and in many cases alternate strategies abound. When you find an error, or if anything is unclear, let me know and I will fix it.
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!