**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