The following exercises use Singular.

• Read the help file for Examples of ring declarations. You can access the help either on the Singular homepage (it's section 3.3.1 of the manual), or by typing help; in Singular (provided you installed help files). What command would you use to define r to be the ring Q[x]?
• Read the help file for the command gcd (you can type help gcd; if you installed the help files) and do problem §1.5#8 from the book (hint: there's a very similar example on page 45 of the book). Your answer should be in the form of a printout from a Singular session.
• Do problem §1.5#9 from the book.

The following exercise is about well orderings. Which of the following order relations are well orderings? If your answer is "no", explain why.

• The subset [0,1] of the real line, with the usual ordering.
• The subset {-1/n | n is a natural number} of the real line, with the usual ordering.