A. Student

Math 124 Problem Set 7

1. D=-155 There are four elements:
By the structure theorem, is isomorphic to either x or . It is easy to verify that is the identity. From this we find that has order 4, so it must be that .
D=-231 There are twelve elements:
Therefore or x. The identity is . Both and have order 6, which is impossible in , so x.
D=-660 There are eight elements:
The first element is the identity, and all others have order 2. Therefore xx.
D=-12104 There are forty-eight elements: (listed in an email from Professor Stein). By the structure theorem, , x, or x. The identity element is , and using it we find two elements of order four: and , eliminating everything but x.
D=-10015 There are fifty-four elements (listed in an email from Professor Stein). Therefore x or . The identity is ; from this we find two elements with order 9: and . Therefore the group cannot be , so x.
2. The three graphs are on the next page, plotted in MAPLE.
3. Differentiating implicitly, the slope of the tangent at is . At , the slope is , and the tangent line has equation . Substituting into the relation , we have , which simplifies to the polynomial

This polynomial has a double root at , so it factors into , giving a rational root with . Therefore is a rational solution to the original equation.