MAS400: Exercises 2

Throughout this sheet F will denote an arbitrary field unless otherwise stated.

1. Show that ℤ[x] has ideals that are not generated by a single element. Thus ℤ[x] is not a principal ideal domain, even though it is a unique factorisation domain.

2. Write, using pseudo-code, the univariate division algorithm (that is the algorithm that given f, g ∈ F [x] with g ≠ 0 will return q, r ∈ F [x] such that f = qg + r and deg r < deg g).

3. Rewrite the Euclidean Algorithm so that only a bounded number of variables is used. (These variables should be elements of F [x], and not such things as an arbitrarily long sequence of elements of F [x].)

4. Show that the only monomial order on ℕ is x^{0} < x^{1} < x^{2} < ⋯ < x^{i} < x^{i+1} < · · ·.

5. Show that a totally ordered set is well ordered if and only if it has no infinite descending chains.

6. Any exercises that may be embedded in my lecture notes.