Math 562
Homework 5
Spring 2012
Drew Armstrong
Problems that are all connected:
We say that a polynomial f (x1 , x2 , . . . , xn ) F [x1 , x2 , . . . , xn ] is symmetric if for every permutation of cfw_1, 2, . . . , n we have
f = f (x1 , x2 , . . . , xn )
Math 562
Homework 3
Spring 2012
Drew Armstrong
Problems on Number Theory
The rst problem substitutes for the proof of FLT for exponent 3, which was too hard.
1. Prove that the equation y 3 = x2 + 2 has exactly two integer solutions: (x, y) = (5, 3).
(a) I
Math 562
Homework 2
Spring 2012
Drew Armstrong
Problems on General Rings
1. We say that a general ring R is (right) Artinian if every descending chain of (right) ideals
terminates. That is, given ideals R I1 I2 there exists some k 1 such that Ik = Ik+1 =
Math 562
Homework 1
Spring 2012
Drew Armstrong
1. Let R be a ring. We say that a R is nilpotent if an = 0 for some n. If a is nilpotent,
prove that 1 + a and 1 a are units (i.e. invertible).
2. Let I R be an ideal. Prove that I = R if and only if I contai
Math 562
Homework 6
Spring 2012
Drew Armstrong
Problems on Galois Connections
Let S, T be sets and let R S T be a relation (we will write aRb to denote the statement
(a, b) R). For all subsets A S and B T let us write
A := cfw_t T : aRt a A T,
B := cfw_s
Math 562
Exam 3
Friday, April 27
Drew Armstrong
There are 3 problems and 4 pages. This is a closed book test. Any student caught cheating
will receive a score of zero.
1. Let F K be a nite-dimensional (hence algebraic) extension of characteristic 0 elds.
Math 562
Exam 1 Fri Feb 17
Spring 2012
Drew Armstrong
There are 4 problems and 4 pages. This is a closed book test. Any student caught cheating will
receive a score of zero.
1. Let R be an integral domain and consider a, b, c, p R.
(a) [1 point] If a = 0,
Math 562
Homework 1
Spring 2012
Drew Armstrong
1. Let R be a ring. We say that a R is nilpotent if an = 0 for some n. If a is nilpotent, prove that
1 + a and 1 a are units (i.e. invertible).
Proof. Recall that in any ring we have (a)(b) = (ab) (see HW 3.7
Math 562
Homework 2
Spring 2012
Drew Armstrong
Problems on General Rings
1. We say that a general ring R is (right) Artinian if every descending chain of (right) ideals
terminates. That is, given ideals R I1 I2 there exists some k 1 such that Ik = Ik+1 =
Math 562
Homework 3
Spring 2012
Drew Armstrong
Problems on Number Theory
The rst problem substitutes for the proof of FLT(3), which was too hard.
1. Prove that the equation y 3 = x2 + 2 has exactly two integer solutions: (x, y ) = (5, 3).
(a) If y 3 = x2
Math 562
Homework 4
Spring 2012
Drew Armstrong
Problems on Rings
1. We say that an ideal I R is prime if for all a, b R, ab I implies that a I or b I .
(a) Prove that I R is prime if and only if R/I is an integral domain.
(b) Prove that every maximal idea
Math 562
Homework 5
Spring 2012
Drew Armstrong
Problems that are all connected:
We say that a polynomial f (x1 , x2 , . . . , xn ) F [x1 , x2 , . . . , xn ] is symmetric if for every permutation
of cfw_1, 2, . . . , n we have
f = f (x1 , x2 , . . . , xn
Math 562
Homework 6
Spring 2012
Drew Armstrong
Problems on Galois Connections
Let S, T be sets and let R S T be a relation (we will write aRb to denote the statement
(a, b) R). For all subsets A S and B T let us write
A := cfw_t T : aRt a A T,
B := cfw_s
Math 562
Homework 4
Spring 2012
Drew Armstrong
Problems on Rings
1. We say that an ideal I R is prime if for all a, b R, ab I implies that a I or b I.
(a) Prove that I R is prime if and only if R/I is an integral domain.
(b) Prove that every maximal ideal
Math 562
Wed Mar 28: Exam 2
Spring 2012
Drew Armstrong
There are 3 problems and 4 pages. This is a closed book test. Any student caught cheating
will receive a score of zero.
1. Let R be a commutative ring with 1 and let I R be a maximal ideal.
(a) Given