Next: About this document ...
Math 124 Final Examination
Due Sunday 12 January 2003 by 5pm
William Stein
Date: Math 124 HARVARD UNIVERSITY Fall 2002
This is the Fall 2002 Math 124 take-home final
examination. You may not discuss the problems with anyone. You
are allowed to look at books, course notes, web pages, and use a
computer (MAGMA, Maple, etc.), but you must acknowledge any sources that you
use. If you find the complete solution to one of these problems in a
book, then good job--you are allowed to copy it.
For your convenience, the complete course notes are available as
a single file at
http://modular.fas.harvard.edu/edu/Fall2002/124/stein/.
All problems are worth the same number of points (e.g., problems 2 and
3 are worth the same number of points). Choose and do EXACTLY 8 of the following problems (e.g., all parts of
problems 1, 2, 3, 5, 6, 10, 11, 12 ). Choose wisely; some
problems might be easy, others difficult open problems, and
some might ask you to prove something that is false (say what is wrong
with the problem for full credit). At least eight are not open
problems! Clearly indicate which problem you are attempting and which
you are omiting.
If you have trouble getting into the math department to hand in your
exam, call my office phone (617-495-1790)
or my mobile phone (617-308-0144) so I can open the door.
- The usual Euler function is a map
. Fix a prime number , and define
a polynomial analogue of Euler's function
by
(let ).
Thus, e.g., if then
.
Say that polynomials
are coprime if
that is, and have
no common roots in
.
- Prove that is multiplicative, in the sense that
if and are coprime, then
- The Euler function does not satisfy
for all integers ; for example,
.
Does satisfy
for all polynomials and ? Give a proof or
counterexample.
Write an insightful review of the book Uncle Petros and
Goldbach's Conjecture. (Your intended audience is a ``typical
Harvard undergraduate with greater than usual interest in
mathematics'', and your review should be at least one page
long.)
- Characterize the positive integers such that
is a field.
- Characterize the positive integers such that
is cyclic.
- Characterize the positive integers such that
is a sum of two rational squares.
- Characterize the positive integers such that
is a sum of two rational squares.
At a conference at the American Institute of Mathematics, Victor
Rotger from Barcelona asked me a question about primes. Call a prime
number Victor if for every prime
with
, we have
(that's the quadratic residue
symbol). Victor's Question: Are there infinitely many Victor primes?
Do numerical computations and formulate an intelligent response to
Victor's question. (You don't have to prove anything to get full
credit on this problem; just compute and give a reasonably intelligent
interpretation of what you find.)
- Factor the integer
as a product with .
- Factor the integer
as a product with , , integers . You may use that
and
where is the sum of the divisors of .
- Find a rational number with
such that
- Find three distinct solutions to
with positive integers.
- Let
be a quadratic form
whose discriminant is a perfect square, possibly 0.
Show that factors as
.
- Let
be a quadratic form.
Show that there exists not both zero such that
if and only if the discriminant of is
a perfect square.
- Divide the following set of
binary quadratic forms up into equivalence classes
modulo the action of
:
- Compute the discriminant of the ring of integers of
.
Consider a right triangle the lengths of whose sides are integers.
Prove that the area cannot be a perfect square.
Assume the truth of Fermat's last theorem. Deduce that
there is no right triangle with rational side lengths and area .
- Does
have a solution in the integers?
- Does
have a solution in the rational numbers?
- Does
have any nontivial rational
solution?
- Does
have any nontrivial rational
solutions?
- Show that there are no positive integers and such
that and are both perfect squares.
- Suppose is a square-free integer, let
and let be the ring of integers in . Prove that the group
of units in has order .
- For which squarefree integers is the
prime ramified in
?
(We say that ramifies in
if
for
some prime of .)
Some elliptic curve questions:
- Show that the number of pairs with
such that
is exactly .
- Suppose and
are such that
with
. Let be the unique solution to . Show
that is a repeated root of the polynomial .
- Suppose that the polynomial
has no
repeated roots, and let be the elliptic curve over
defined
by . Show that
Next: About this document ...
William A Stein
2003-01-08