**189-235A:** Basic Algebra I

## Assignment 4

## 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|*.

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)*.