Math 772
Spring 2002
Exam 1
Due
Wednesday February 20, 2002.
a)
Show
that the smaller ring enjoys the property that given any element , there is a unit such that (so in a certain
sense, every element of the larger ring is “almost” in the smaller ring).
b)
Despite
this unreasonable niceness, show that is not a unique
factorization domain.
c)
Show
that is, however, an HFD
(any two factorizations of the same nonzero nonunit have the same length).
a)
Show
that the equation can be translated to
the equation by an appropriate
change of variables.
b)
Assume
a solution of the form Show that the
(reasonable?) assumption that the other two solution are of the form and its conjugate
reduces the equation to a quadratic.
c)
Derive
the cubic formula (and astound yourself with some identities).