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Homework Assignment 1
Due Wednesday October 2
William Stein
Date: Math 124 HARVARD UNIVERSITY Fall 2002
Instructions: Please work with others, and
acknowledge who you work with in your write up. If you can do a problem
using a computer please do, but describe how you use the computer. For
more practice you can do the problems in the book.
There are six problems.
- (1 point each) Compute the following gcd's using
the Euclidean algorithm (show the steps):
and
- (2 points)
Use the Euclidean algorithm to find integers
such that
- (2 points each)
Let
be the ring of elements of the
form
such that
. An nonzero non-unit
in is irreducible if the only divisors of
are of the form with a unit. Also, the norm of
is
- Find the units in .
- Prove that if then
.
- Show that is irreducible in the ring
.
[Hint: If
and take norms.]
- Show that
is irreducible in
.
[Hint: If
and take norms.]
- (4 points)
Find the second smallest positive integer such that
- (5 points)
Suppose that
, and let
be lifts of , respectively.
Prove that
doesn't
depend on the choice of
.
- (2 points) Prove that
is the number of units
in
.
- (4 points) Prove that is multiplicative as follows. Show that
the natural map
is
an injective map of rings, hence bijective by counting, then
look at unit groups.
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William A Stein
2002-09-26