Math 461
Homework 2
Spring 2015
Drew Armstrong
1. Let F be a eld and consider two polynomials f (x), g(x) F[x].
(a) If f (x) and g(x) are both nonzero, prove that deg(f g) = deg(f ) + deg(g).
(b) How should you dene the degree of the zero polynomial so th
Math 461
Homework 1 Solutions
Spring 2015
Drew Armstrong
1. In al-Khwarizmis solution of quadratic equations he needed to solve the following geometric
problem. Consider a line segment AB. Let C be its midpoint and let D be any other point
on the segment.
Math 461
Homework 3
1. De
(a)
(b)
(c)
Spring 2015
Drew Armstrong
Moivres Theorem.
Use de Moivres Theorem to express cos(2) as a polynomial in cos().
Solve this polynomial to obtain a formula for cos() in terms of cos(2).
Use the formula from (b) to nd the
Math 461
Homework 5
Spring 2015
Drew Armstrong
1. Equivalence mod n. Let S be a set. For each pair (x, y) S 2 we dene x y to be
either true or false, and we will usually write x y as shorthand for x y is true. We say
that is an equivalence relation if
x
Math 461
Homework 6
Spring 2015
Drew Armstrong
1. Symmetric Functions. Consider the elementary symmetric functions
e1 = r + s + t
e2 = rs + rt + st
e3 = rst.
They are called elementary because every other symmetric function can be expressed in terms
of th
Math 461
Homework 6
Spring 2015
Drew Armstrong
1. Symmetric Functions. Consider the elementary symmetric functions
e1 = r + s + t
e2 = rs + rt + st
e3 = rst.
They are called elementary because every other symmetric function can be expressed in terms
of th
Math 461
Homework 4
Spring 2015
Drew Armstrong
1. Dierence of Like Powers. Let n be a positive integer and dene := e2i/n . Prove
that for all numbers a and b we have
an bn = (a b)(a b)(a 2 b) (a n1 b).
Proof 1: First note that the formula is true for b =
Math 461
Homework 5
Spring 2015
Drew Armstrong
1. Equivalence mod n. Let S be a set. For each pair (x, y) S 2 we dene x y to be
either true or false, and we will usually write x y as shorthand for x y is true. We say
that is an equivalence relation if
x
Math 461
Homework 3
1. De
(a)
(b)
(c)
Spring 2015
Drew Armstrong
Moivres Theorem.
Use de Moivres Theorem to express cos(2) as a polynomial in cos().
Solve this polynomial to obtain a formula for cos() in terms of cos(2).
Use the formula from (b) to nd the
Math 461
Homework 2
Spring 2015
Drew Armstrong
1. Let F be a eld and consider two polynomials f (x), g(x) F[x].
(a) If f (x) and g(x) are both nonzero, prove that deg(f g) = deg(f ) + deg(g).
(b) How should you dene the degree of the zero polynomial so th
Math 461
Homework 1
Spring 2015
Drew Armstrong
1. In al-Khwarizmis solution of quadratic equations he needed to solve the following geometric
problem. Consider a line segment AB. Let C be its midpoint and let D be any other point
on the segment. Construct
Math 461
Impossible Constructions
Spring 2015
Drew Armstrong
In class we proved that 2 is not a rational number (the edge and diagonal of a square
are incommensurable). This crisis of incommensurables forced the Greeks to base their
mathematics on the con
Math 461
Exam 1 (Wed Feb 18)
Spring 2015
Drew Armstrong
There are 4 problems, each with 3 parts. Each part is worth 2 points, for a total of 24 points.
If any two exams are submitted with copied answers then both exams will receive 0 points.
1. Division W
Math 461
The Cubic Formula
Spring 2015
Drew Armstrong
The full cubic formula is too complicated to write on the board, so Ive typed it here.
Recall that the solution to the depressed cubic
x3 + px + q = 0
is given by Cardanos formula
p
q
q 2
+
+
2
2
3
Now
Math 461
Exam 2 (Wed Mar 25)
Spring 2015
Drew Armstrong
There are 3 problems with 12 parts. Each part is worth 2 points for a total of 24 points. If
two exams are submitted with copied answers then both exams will receive 0 points.
1. Complex Numbers.
(a)
Math 461
Rings and Fields
Spring 2015
Drew Armstrong
I will avoid stating the formal denition of rings and elds in the lecture, but here it is
in case you want to know.
Denition of a Ring. A ring is a set R together with two binary operations
+:RRR
and
:R