Mathematics 501
Homework (due Oct 25!)
A. Hulpke
36) The cyclotomic polynomial Φn (x) of order n is defined as the minimal polynomial of
2π i
ξn := e n (i.e. the monic rational polynomial of smallest degree that has ξn as a root). In
Algebra one shows that Φn (x) = ∏ (x − ξni ).
gcd(i,n)=1
a) Show that x − 1 = ∏ Φd (x).
d |n
b) Using a multiplicative version of Möbius inversion determine a formula for Φn (x).
c∗ ) Write a function to calculate Φ(n) for a given value of n1 . Using this function examine
the coefficients of Φ(n) for several values of n. Do coefficients different from 0, ±1 arise?
n
37) A truncated icosahedron (sometimes known as “soccer ball”)
has faces that are pentagons and hexagonsa . Every hexagon borders
three hexagons and three pentagons, every pentagon is surrounded
by hexagons. (Thus every vertex lies on exactly one pentagon.)
Using Euler’s polyhedron theorem, determine the number of
pentagons and hexagons.
a
why is it not possible to only use hexagons?
38) A platonic solid is a polyhedron in which all faces are congruent regular convex
polygons such that the same number of polygons meet at each corner. Using Euler’s
polyhedron theorem, show that there are exactly 5 platonic solids.
39) a) Determine µ(0, 1) = µ(h1i , D8 ) for the lattice of subgroups
of D8 , the dihedral group of order 8, given on the side.
b) Consider the topological space given by rooms, corridor
and stairs in the Weber building occupied by the Mathematics
department (excluding elevator, bathroomsa , custodial rooms, the
ACNS/Primes rooms, WB202 and the PACe rooms). Assuming all
inside doors are open (and the rooms empty), what is the Euler
characteristic of this space?
a
Since nobody can be expected to investigate male and female bathrooms
40) Consider a tableau of shape (n, n). Define a sequence ak (1 ≤ k ≤ 2n) by setting ak := i
if k is in row i of the tableau, i = 1, 2. Use this to show that the number of tableaux of this
shape is given by the Catalan Number Cn+1 . (cf. Problem 16!)
1
In GAP, for example, you would want to set x:=X(Rationals,"x"); to define a polynomial variable,
then use the functions DivisorsInt and MobiusMu.
41) For2 an integer m define the “root of discriminant” function as:
∆(x1 , . . . , xm ) :=
∏
(xi − x j )
1≤i< j≤m
We now define a polynomial g on n + 1 variables by
g(x1 , x2 , . . . , xn , y) = x1 ∆(x1 + y, x2 , . . . , xn ) + x2 ∆(x1 , x2 + y, x3 , . . . , xn )
+ · · · + xn ∆(x1 , x2 , x3 , . . . , xn + y)
a) Show that g is homogeneous (all terms have the same total degree) and that it is
antisymmetric in the xi ’s (i.e. exchanging xi and x j changes the sign of g). Conclude that
there must be a polynomial h such that
g(x1 , x2 , . . . , xn , y) = h(x1 , . . . , xn , y) · ∆(x1 , . . . , xn )
b) Show that the polynomial h defined this way must have the form
h(x1 , . . . , xn , y) = a(x1 + · · · + xn ) + by
c) Determine the value of thevariable
a by setting y = 0.
n
xi
.
=
d) Show that ∑ ∑
2
i j6=i xi − x j
∂ g e) Calculate the value of the variable b, using that b =
.
∂ y y=0
f) Conclude that
n
g(x1 , x2 , . . . , xn , y) = x1 + x2 + · · · + xn +
y · ∆(x1 , . . . , xn )
2
g) Conclude by setting y = −1 that formula (34) in the handout fulfills the recursion (32).
42) Read sections 12.9, 13.2 and 13.3 in the book.
Problems marked with ∗ are bonus problems for extra credit.
Note: I will be out of town October 16-20, there will be no lectures and office hours in
that week. Thus this homework is due only in two weeks time.
2
This is problem 17 from TACP, 5.1.4