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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).