MATH 4000
Spring 2014 (Azo )
Notes to Homework Problems
These are more in the nature of hints than complete proofs. In particular, comments about bonus problems
(marked by *) are particularly brief.
Assignment # 1, Problems from Section 1.1
1.1.4 Inductio
2.3.10 Applying 6d) and substituting sin2 = 1 cos2 yields cos 4 = 8 cos4 8 cos2 + 1. Substitute
= to see that x = cos satises the equation 8x4 8x2 + 1 = 0. The quadratic formula yields
8
8
x2 = 24 2 . We choose the positive sign because the cosine funct
2.2.5 Take a =
2 and b = 2. Then a + b and ab are both rational.
2.2.6a Assuming a2 = 3b2 with a, b relatively prime, reach the contradiction that a, b are both divisible
by 3. Some of you gave a creative alternae argument based on parity, but the precedi
*1.1.19 limn an = 0. To see this, note that the rst 2n+1 rows of Pascals triangle contains three
copies of the rst 2n rows of the triangle surrounding a large triangle of zeros. It follows that a(2n+1 ) is
approximately equal to 3 a(2n ) and limn a(2n ) =
For the desired counterexample, choose a, b to be invertible matrices which do not commute.
1.4.10 a = a(bc) = (ab)c = c.
*1.4.13 The only elements of Zp which are their own inverses are 1 and 1 = p 1.
1.4.19b f R is a zero divisor i it is not identically
Assignment # 3, Problems from Section 1.3
1.3.1 For reexivity, dont assume what youre trying to prove. Rather let a Z, note that m|a a = 0
so a a mod m by denition.
1.3.2b 10 1 mod 10 implies 10i 1 for 0 i k whence
k
k
10i ai
N :=
i=1
Thus 3|N i N 0 mod
a) rational root test
c) factor by inspection
e) factor by inspection (quadratic in x2 )
g) reduce modulo 5
i) factor by inspection
3.3.ace In e), rst multiple by 5 to clear the factions
3.3.4 In b), use Eisenstein with p = 2.
In c), rt consider the case
3.1.20 Follow the procedures of Section 1.3.
*3.1.9c f (2) = f (5) = 0.
*3.1.16 Reduce to the case where g is monic.
*3.1.19 Assume only that R is an integral domain and and note that x divides a polynomial h(x) R[x]
i h(0) = 0. It follows that if x divid
MATH 4000/6000
Fall, 2012
PROBLEM SET #3
DUE Wednesday, September 5, 2012.
(Note the one-time change because of the Labor Day holiday.)
Problems to work but not hand in :
Chapter 1, 3: #7, 9, 12, 13, 19, 20a,d,e, 21a,c.
Problems to turn in :
A. Prove that
MATH 4000/6000
PROBLEM SET #2
Fall, 2012
T. Shifrin
DUE Monday, August 27, 2012.
Problems to work but not hand in :
Chapter 1, 2: #8, 9, 10, 16.
Chapter 1, 3: #1, 2, 3, 4, 5.
Problems to turn in :
Chapter 1, 2: #7, 13.
A. Prove: gcd(a, b) = gcd(a + kb, b)
MATH 4000/6000
Fall, 2012
PROBLEM SET #14
DUE Monday, November 26, 2012.
Problems to work but not hand in :
Chapter 4, 3: #1, 2, 3, 6, 8.
Problems to turn in :
A. (2) Find the multiplicative inverse of x3 + 2x + 1 in F [x]/ x2 x 1
for
(i) F = Q
(ii) F = Z
MATH 4000/6000
Fall, 2012
PROBLEM SET #6
T. Shifrin
DUE Monday, September 24, 2012.
Problems to work but not hand in :
Chapter 2, 3: #1, 2, 3, 9a,c, 15a, 22.
Chapter 2, 4: #1a,c,d, 2b.
Problems to turn in :
A. (4) Express the following roots in closed for