Math 494 (Winter 2013). Solutions for Problem Set 1
1.1. Morandi, 1.2. The inclusion F K allows us to view F as an F -vector subspace of
K . We have dimF (F ) = 1. A basic fact of linear algebra is that if V is a (possibly innite
dimensional) vector space
Math 494 (Winter 2013). Honors Algebra II
Problem Set 3. Due Friday, February 1
3.1. Recall that in class, we agreed to take the following as our main denition of a
Galois extension. A nite extension of elds L K is said to be Galois if the order of
the gr
Math 494 (Winter 2013). Honors Algebra II
Problem Set 2. Due Friday, January 25
2.1. Let K be a eld. A polynomial F (x, y ) K [x, y ] in two variables over K is said
to be symmetric if F (x, y ) = F (y, x). Prove that if F (x, y ) K [x, y ] is a symmetric
Math 494 (Winter 2013). Honors Algebra II
Problem Set 1. Due Friday, January 18
Reading 1. Read Chapter 1 of Morandis book.
Reading 2. Read Chapter 1 of the following notes:
http:/www.math.tifr.res.in/~publ/pamphlets/galoistheory.pdf
This will be mostly a
Math 494 (Winter 2013). Honors Algebra II
Problem Set 5. Due Friday, February 15
The goal of the next three exercises is to complete the proof of
Theorem 5.1. Let L K be a nite eld extension. The following are equivalent:
(i) L is normal over K ;
(ii) [L
Math 494 (Winter 2013). Honors Algebra II
Problem Set 7. Due Friday, March 15
Problems. Solve the following exercises from Chapter 2 of M. Reids Undergraduate
Commutative Algebra: 2.1, 2.2, 2.6, 2.7, 2.8, 2.9, 2.10, 2.12, 2.13, 2.15.
Remarks 7.1. (1) Dont
Math 494 (Winter 2013). Honors Algebra II
Problem Set 11. Due Friday, April 12
11.1. If k is a eld and S k [X1 , . . . , Xn ] is a subset, the denition of V (S ) k n
given in class (see also problem set 10) makes sense even if k is not algebraically close
Math 494 (Winter 2013). Honors Algebra II
Problem Set 10. Due Friday, April 5
10.0. Read Chapter 5 of M. Reids Undergraduate Commutative Algebra.
Denition 10.1. A topological space X is called irreducible if there do not exist closed
subsets Y1 , Y2 X suc
Math 494 (Winter 2013). Honors Algebra II
Problem Set 9. Due Friday, March 29
Problems. Solve the following exercises from Chapter 4 of M. Reids Undergraduate
Commutative Algebra: 4.1, 4.4, 4.5, 4.6 (assume that the eld k has characteristic = 2!),
4.7, 4.
Math 494 (Winter 2013). Honors Algebra II
Problem Set 12. Due Friday, April 19
12.0. Read Chapter 6 of M. Reids Undergraduate Commutative Algebra.
12.1. Let A be a commutative ring and S A a multiplicative subset. Show that if A is
Noetherian, then so is
Math 494 (Winter 2013). Honors Algebra II
Problem Set 4. Due Friday, February 8
4.1. Use the results of the previous problem set (including the extra credit problems) to
5
10
15
6
19
compute the following Legendre symbols:
,
,
,
,
.
37
37
37
23
43
4.2. Pr
Math 494 (Winter 2013). Solutions for Problem Set 12
12.1. For each ideal I A, we have the ideal S 1 I S 1 A, and the operation I S 1 I
is visibly inclusion-preserving. In addition, it is proved in Corollary 6.3(ii) of Reids book
Undergraduate Commutative
Math 494 (Winter 2013). Solutions for Problem Set 11
A
11.1. I will omit the pictures since Im too lazy to gure out how to draw them in L TEX.
Here is a description of what they look like: (a) is the union of the two lines given by
X1 = X2 and X1 = X2 ; (
Math 494 (Winter 2013). Solutions for Problem Set 2
2.1. Saying that F (x, y ) K [x, y ] is symmetric is the same as saying that for any integers
m > n 0, the coecients of xm y n and of xn y m appearing in F (x, y ) are equal to each
other. So if F (x, y
Math 494 (Winter 2013). Solutions for Problem Set 3
n
3.1. We use the notation of the hint. From Math 493, we have xq = x for all x L, so
the order of the automorphism (x) = xq in AutK (L) is at most n. On the other hand,
m
if an integer m 1 is such that
Math 494 (Winter 2013). Solutions for Problem Set 4
4.1.
All Legendre symbol calculations can be done essentially mechanically using the following
m
n
principles:
=
if p is a prime and m n mod p (obvious from the denition);
p
p
m
n
mn
=
for any prime p (f
Math 494 (Winter 2013). Solutions for Problem Set 5
5.1. Assume that L K is a nite normal extension, g (x) K [x] is irreducible and L is a root
of g . Choose an algebraically closed extension E L; in particular, we may view E as an algebraically
closed ex
Math 494 (Winter 2013). Solutions for Problem Set 6
6.1. Reid, 1.1. Let A = Z, I = 2Z and J = 3Z. Then I J is the set of integers that
are divisible by either 2 or 3, and is not an ideal, because it is not even closed under
addition: 2 + 3 = 5. The smalle
Math 494 (Winter 2013). Solutions for Problem Set 7
7.1. Reid, 2.1. Suppose that 0 = f A and A[1/f ] is a nite A-module. Say x1 , . . . , xn
are elements that generate A[1/f ] as an A-module. By denition, every element of A[1/f ]
can be expressed as a (ni
Math 494 (Winter 2013). Solutions for Problem Set 8
8.1. Reid, 3.1. This is very false. For example, let A be any commutative integral
domain, which is not Noetherian as a ring (for instance, A can be a polynomial ring over
a eld in countably many indeter
Math 494 (Winter 2013). Solutions for Problem Set 9
9.1. Reid, 4.1. (a) Given any f (X ) k [X ], we can separate the even and odd powers
of X appearing in f and write f (X ) = g (X ) + X h(X ), where g, h are polynomials
that only involve X 2 , i.e., g (X
Math 494 (Winter 2013). Solutions for Problem Set 10
10.1. It is clear that a topological space consisting of one point is irreducible. We claim that
the only irreducible subsets of Rn (with the usual topology) are the singleton sets. In fact,
something r
Math 494 Homework Set 1
Due January 20 Wednesday (Monday: MLK Day, no class)
Homework is due on Wednesday at the beginning of class. Late homework is not accepted. I
encourage you to work with others on homework problems, but you must write up your own so