OLD ALGEBRA MIDTERM
No books or notes. In multipart problems you may (and should) assume
earlier parts when working on later parts. Max score: 105
1. (15) Set = e2i/15 = cos(2/15) + i sin(2/15).
(a) (5) Mark clearly on the complex plane where is.
(b) (5)
ALGEBRA MIDTERM
1. (20) Set = e2i/18 = cos(2/18) + i sin(2/18).
(a) (5) Mark clearly on the complex plane where is.
(b) (5) Do and 14 have the same minimal polynomial over Q?
Why?
(c) (10) Give as good an upper bound on [Q() : Q] as you can.
2. (10) Find
Algebra V63.0344
Assignment 6
Due, Friday, Mar 15 in Recitation
May my mind stroll about hungry and fearless
and thirsty and supple
and even if its Sunday may I be wrong
for whenever men are right, they are not young
e.e. cummings
1. Let = a + bi C . Let
Algebra V63.0344
Assignment 5
Due, Friday, Mar 1 in Recitation
The universe is not only queerer than we suppose but queerer
than we can suppose.
J.B.S. Haldane
1. Let C be a root of x3 + x +3. (This cubic has no special properties.)
Writs i in the form a
Algebra
Presidents Day Entertainments
The one important thing I have learned over the years is the
dierence between taking ones work seriously and taking ones
self seriously. The rst is imperative and the second is disastrous.
Margot Fonteyn
1. Sing Sher
Algebra V63.0344
Assignment 3
Due, Friday, Feb 15 in Recitation
It is not knowledge, but the act of learning, not possession but
the act of getting there, which grants the greatest enjoyment.
When I have claried and exhausted a subject, then I turn away
f
Algebra V63.0344
Assignment 2
Due, Friday, Feb 8 in Recitation
Math is natural. Nobody could have invented the mathematical
universe. It was there, waiting to be discovered, and its crazy;
its bizarre. John Conway
1. In Z7 [x] let f (x) = 2x5 + 3x4 + 4x +
Algebra V63.0344
Assignment 1
Due, Friday, Feb 1 in Recitation
I have no home. The world is my home. Paul Erds
o
Prof. Spencers email: spencer@cims.nyu.edu. For despamication, please
make subject matter Algebra.
Special Note: There will be recitation the
Algebra V63.0344
Notes: 3/13/13
Continuing with Finite Fields
Theorem: Let |F| = pN
(i) F, Zp () = F
(ii) f (x) Zp [x], irreducible inZp [x], with F Zp [x]/(f (x)
=
Proof:
(i) Pick a generator of F . Clearly, F Zp [], and because is algebraic over F of de
1
Recitation
In this rst recitation we discuss some basic results about complex numbers
and related polynomial rings.
Complex Numbers
As a set, we can describe the complex numbers as tuples of real numbers:
C = cfw_(a, b) | a, b R
We can make the complex