189-235A: Basic Algebra I
Due: Wednesday, October 1.
1 Page 149, 3.6, 3.7, 3.11.
2. Page 155, 3.18, 3.20, 3.23.
3. Page 161, 3.28.
4. Let R be
the ring of Gaussian integers consisting of all elements
of the form m+ni, where m and n are regular integers.
a) Define the norm of an element of R by the rule
|m+ni| = m^2+n^2. Show that, if a divides b in
R, then |a| divides |b|.
Show that R has a
division algorithm: namely, for all a and b in R,
there exist q and r in R such that
a=bq+r and |r| <|b|.
c) A gcd of a and b is an element of R which divides both
a and b and whose norm is maximal. Show that, if c and
d are two gcd's of a and b, then their ratio is a power of i.
d) Using part b), describe an algorithm to compute a gcd of two elements of
R. Use this algorithm to compute gcd(7+9i,7-4i).