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MTH5101 Theory: Guide to Coursework 4A

Note: These are by no means complete solutions!

1. (a) Write down the definition of an integral . Give an example of an , and an example of a domain that is not an integral domain. An integral domain is a domain containing no zero-divisors, that is, a domain such that if a · b = 0 then either a = 0 or b = 0. An example of an integral domain is the ring Z, and an example of a domain that is not an integral domain is Z/6Z (as [2]6 · [3]6 = [0]6).

2. Consider the domain R = Z/14Z = {[0]14, [1]14,..., [13]14}.

(a) List all the zero-divisors of R, and explain why they are zero-divisors. The zero-divisors correspond to the that are not coprime to 14:

[2]14, [4]14, [6]14, [7]14, [8]14, [10]14, [12]14.

You should check that each of them can be multiplied by another (pos- sibly more than one) non- to get [0]14. (b) List all the units of R, and their corresponding inverses. The units correspond to the integers that are coprime to 14:

[1]14, [3]14, [5]14, [9]14, [11]14, [13]14.

Again, make sure to know for each of them which is the (unique) element they need to be multiplied by to get [1]14. (c) Is R an integral domain? Is R a field? The domain R has zero-divisors, so it is not an integral domain. It is also not a field, because there are non-zero elements that are not units.

3. Let R = Z/4Z, and consider the ring R[x] of polynomials with coefficients in R.

(a) Is R[x] a domain? Explain. Since R is a domain, R[x] is also a domain (a with identity). For example, what is the identity of R[x]? (b) Is R[x] an integral domain? Explain. The ring R has zero-divisors, and this leads to many zero-divisors in R[x]. For example, ([2]4 x) · ([2]4 + [2]4 x) = 0 (the zero polynomial), but none of the factors is the zero polynomial. 2 (c) Is the element [2]4 + [0]4 x + [2]4 x ∈ R[x] a zero-divisor? It is. Can you find another non-zero polynomial which multiplied by it is equal to 0?

(d) Is the element [2]4 + [1]4 x ∈ R[x] a zero-divisor? It is not. To see this, think about what the leading coefficient (the coefficient with the highest power of x) is after multiplying by another polynomial. 2 (e) Is the element [2]4 + [3]4 x + [2]4 x ∈ R[x] a ? It is not. Think about the constant coefficient after multiplying by another polynomial.

(f) Is the element [3]4 + [2]4 x ∈ R[x] a unit? It is! Can you find its inverse?