The first assignment is a review of results that you should have
seen in Math 1530. It is material that we will be using
in Math 1560. You should first try to prove the theorems on your own,
but if you get stuck, then you can look them up in an algebra textbook
and rewrite the proofs in your own words. But be sure that you
understand the proofs, as well as the statements of the theorems.
Problem A.1
Let G be a finite group, let H be a
subgroup of G, and let g ∈ G
be an element of G.
(a)
Recall that the order of G is the number of elements
in G, and similarly for H. What is the definition of the
order of the element g?
(b)
Lagrange's Theorem (Part 1):
Prove that the order of H divides the order of G
(c)
Lagrange's Theorem (Part 2):
Prove that the order of g divides the order of G
Problem A.2
Let R be a ring.
(a)
What is the definition of an ideal?
(b)
What is the definition of a maximal ideal?
(c)
Let I be an ideal of R. Prove that the quotient ring
R/I is a field if and only if I is a maximal ideal.
(You may assume that R/I is already a ring, so what you need
to prove is that every non-zero element of R/I has an inverse
if and only if I is a maximal ideal.)
Problem A.3
Let Z be the ring of integers {...-2,-1,0,1,2,...}.
(a)
Prove that every ideal in Z is principal, i.e., generated
by a single element.
(b)
Let m ∈ Z. The ideal generated by m is denoted by
(m) or mZ. Prove that mZ is a maximal ideal if
and only if |m| is prime. Deduce that the quotient ring
Z/mZ is a field if and only if |m| is prime.
Problem A.4
Let F be a field.
(a)
Let f(x) ∈ F[x] be a non-zero polynomial of degree d.
Prove that f(x) has at most d distinct roots in F.
(b)
Give an example of a ring R and a non-zero polynomial
f(x) ∈ R[x] of degree d that has more than d
roots in R.
(c)
Let F be a finite field, i.e., a field with finitely many elements.
Prove that there is a prime p such that pa = 0 for all a ∈ F.
(Here pa means to add a to itself p times.)
(d)
Let F be a finite field. Prove that the number of elements in F
is a power of a prime. (Hint. Let p be the prime from (c),
and prove that F is a finite dimensional vector space over the
field Z/pZ.
Problem Chapter 1, # 9
Suppose that u, v ∈ Z and (u,v) = 1. If u | n
and v | n, show that uv | n. Show that this is false
if (u,v) ≠ 1
Problem Chapter 1, # 16
If (u,v) = 1 and uv = a2, show that both
u and v are squares.
Problem Chapter 1, # 21
Prove that ordp(a+b) ≥ min(ordpa,ordpb),
with equality holding if ordpa ≠ ordpb.
Problem Chapter 1, # 24
Prove the following identities:
(a)
xn–yn
= (x–y)(xn–1+xn–2y+...+yn–1).
(b)
For n odd, xn + yn
= (x+y)
(xn–1–xn–2y+xn–3y2–...+yn–1).
Problem Chapter 1, # 25
If an–1 is prime, show that a=2 and that n is prime.
Primes of the form 2p–1 are called Mersenne Primes.
For example, 23–1=7 and 2^5–1=31.
It is not known if there are infinitely many Mersenne primes.
Problem Chapter 1, # 26
If an+1 is prime, show that a is even and that n is
a power of 2. Primes of the form 22t+1 are
called Fermat Primes. For example,
221+1=5 and 222+1=17.
It is not known if there are infinitely many Fermat primes.
Problem Chapter 1, # 30
Prove that 1/2 + 1/3 + 1/n is not an integer.
Problem Chapter 2, # 6
For a rational number r, let [r] be the largest integer less
than or equal to r, e.g., [1/2] = 0, [2] = 2, [10/3] = 3.
Prove that ordp n! = [n/p] + [n/p2]
+ [n/p3] + ...
Problem Chapter 2, # 9
A function on the integers is said to be multiplicative
if f(ab) = f(a)f(b) whenever (a,b) = 1. Show that
a multiplicative function is completely determined by its values on
prime powers.
Problem Chapter 2, # 10
If f(n) is a multiplicative function, show that the
function g(n) = ∑d | n f(d) is also
multiplicative.
Problem Chapter 2, # 11
Show that φ(n) = n ∑d | n μ(d)/d by
first proving that μ(d)/d is multiplicative and then using
Exercises 9 and 10.
Problem Chapter 2, # 16
Show that ν(n) is odd iff n is a square.
Problem Chapter 2, # 20
Prove that ∏d | n d = nν(n)/2.
Problem Chapter 2, # 25
Consider the function ζ(s) = ∑n=1∞ 1/ns.
ζ(s) is called the Riemann zeta function. It converges for s > 1.
Prove the formal identity (Euler's identity)
ζ(s) = ∏p (1–(1/ps))–1
Problem Chapter 2, # 26(a)
Verify the formal identities
(a)
ζ(s)–1 = ∑n=1∞ μ(n)/ns.
Let F be a finite field of characteristic q that
contains a primitive pth root of unity ζ, and
let τa =
∑t∈Fp
(t|p)ζat be the Gauss sum that we
defined in class. (Here and in Problems B.1 and B.2, we write
∑t∈Fp for the sum
from t = 0 to t = p–1.) Also let τ =
τ1.
Problem B.1
What is the value of the sum
∑a∈Fp τa?
Problem B.2
Prove that the Gauss sum τ is equal to the sum
∑t∈Fp
ζt2. (Hint. Evaluate the sum
∑t∈Fp
(1+(t|p))ζt in two ways.)
Problem B.3
Prove that there is a constant A such that
∑n ≤ x (log n)/n
= (1/2) (log x)2 + A + O((log x)/x).
Problem B.4
Prove that
∑n ≤ x d(n)/n
= (1/2) (log x)2 + 2 γ log x + O(1),
where γ is Euler's constant.
Problems B.5, B.6, and B.7 ask you to prove some properties of the
greatest integer function [x].
Problem B.5
Prove that for all real numbers x, the quantity
[2x] – 2[x] is equal to either 0 or 1.
Problem B.6
Prove that [x] + [x + 1/2] = [2x] for all real numbers x.
More generally, prove that
∑0≤k<n [x + k/n] =
[nx].
Problem B.7
For which integers n ≥ 1 is it true that [√n]
divides n?
A Pythangorean Triple is a triple of positive integers
(a,b,c) satisfying
a2 + b2 = c2.
It is a Primitive Pythangorean Triple
if gcd(a,b,c) = 1.
Problem C.1
(a) Let (a,b,c) be a primitive Pythagorean triple.
Prove the ab ≡ 0 (mod 3).
(b) Formulate and prove a similar result modulo 5.
Problem C.2
Find three (or more) primitive Pythagorean triples that
have the same c value, but different a and b values.
(Switching a and b does not count as different.)
Problem C.3
Describe as precisely as you can all solutions in postive integers to
the equation a2 + b2 =
2c2.
Problem D.1
The elliptic curve E : y2
= x3 + 17 has several points having integer
coordinates, including
P = (–2,3),
Q = (–1,4),
R = (2,–5).
(a) Use the line through P and R to find a new point
on E with rational coordinates.
(b) Use the line through Q and R to find a new point
on E with rational coordinates.
(c) Search for other points on E having integer coordinates. How
many can you find?
Problem D.2
Let P = (a,b) be a point on the elliptic
curve E : y2 = x3
+ k, and let L be the tangent line to E at the
point P.
(a) Find a formula for the other point where L intersects E.
(b) Using the elliptic curve E and the point P from Problem D.1,
apply your formula from (a) to find a new point on E with rational
coordinates.
(c) Using the elliptic curve E and the point Q from
Problem D.1, apply your formula from (a) to find a new point
on E with rational coordinates.
Problem D.3
Let e(1), e(2), e(3), ... be a sequence of
positive integers satisfying
e(n+1) ≥ ne(n)
for all n = 1, 2, 3,... .
(a)
Prove that the real number
∑n ≥ 1 2–e(n)
is transcendental.
(b)
Show that the construction in (a) yields uncountably many
transcendental numbers.
Problem D.4
A real number α is said to be d-approximable if there
are infintely many rational numbers p/q satisfying
|p/q – α| < 1/qd.
Thus for example, Dirichlet's theorem says that every real number is
2-approximable.
We consider the following set of d-approximable numbers:
Ad
= { α ∈ R : 0 ≤ α < 1 and
α is d-approximable }.
Prove that if d > 2, then the set Ad
has measure zero. (If you haven't taken measure theory, this means
the following. Show that for every ε > 0 there is an
infinite sequence
of intervals I1, I2,
I3, I4,
... such that Ad is contained in the union of
the Ik's and such that the sum of the lengths
of all of the Ik's is smaller than ε)
The Bernoulli polynomials
Bk(x) are defined by
TexT/(eT – 1)
= ∑k≥0 Bk(x) Tk/k!
Further, the kth Bernoulli number Bk
is obtained by setting x = 0, i.e., Bk
= Bk(0). A hint for many of these problems
is that it is often easiest to prove formulas for
all Bk(x) simultaneously using the
power series definition.
Problem E.1
Prove that Bk(x) is a monic polynomial
of degree k.
Problem E.2
Prove that Bk(1 – x) =
(–1)k Bk(x).
Problem E.3
Prove that Bk(x + 1)
– Bk(x)
= kxk–1.
Problem E.4
Prove that
(d/dx)Bk(x)
= kBk–1(x).
Problem E.5
Prove that Bk(1)
= Bk for all k ≥ 2
and that Bk = 0 for
all odd k ≥ 3. (Hint:
These can be done directly, but it might be
quicker to use problems E.2 and E.3.)
Problem E.6
Let N ≥ 1 be an integer. Prove that
∑0≤i<N Bk(x
+ i/N)
= N1–k Bk(Nx)
Relations of this sort are called distribution relations.
The gamma function Γ(s) is defined by the integral
∫0∞ e–t
ts–1 dt.
Problem F.1
Prove that the integral defining Γ(s) converges
for s > 0. (Note that for 0 < s < 1, the integral is
improper at both limits of integration.)
Problem F.2
We proved in class that it is possible to extend the definition of
Γ(s) to all real s except for non-positive integers,
where Γ(–n) = ∞. Prove that
(s + n)Γ(s) →
(–1)n/n!
as
s → –n.
Problem F.3
Prove that Γ(1/2) = √π. (Hint:
Write Γ(1/2)2 as a double integral, make a change of
variables to get rid of the square root, then change to polar
coordinates to evaluate.)
Problem G.1
Prove that
∑k≥2 (&zeta(k) – 1)
= 1.
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