Math 567 Number Theory I
Homework 7
Name: Zhaoning Yang
October 28, 2013
Problem 1
(Textbook P.216 #4) Find all ideals in Z + Z 6 that contain 6.
Solution: Let K = Q( 6), we know OK = Z + Z 6 is a Dedekind domain. Note for any A D
an ideal of D, 6 A 6 A.
Math 567 Number Theory I
Homework 5
Name: Zhaoning Yang
October 11, 2013
Problem 1 (Textbook P.138 #3) Let K = Q() be an algebraic number eld of degree n. Let 1 =
, 2 , . . . , n be the conjugates of over Q. Suppose that there are exactly m distinct elds
Math 567 Number Theory I
Homework 2
Name: Zhaoning Yang
September 23, 2013
Problem 1 (Textbook P.71 #3) Let I1 I2 . . . be an ascending chain of ideals in an integral domain
D. Prove that In is an ideal in D.
n=1
Solution:
Proof. Denote I = In and suppose
Math 567 Number Theory I
Final Exam
Name: Zhaoning Yang
December 18, 2013
Problem 1.
Solution:
Compute H(Q( 39).
Let K = Q( 39). Because 39 1(mod4), we know OK = Z + Z
1+ 39
2
and d(K) =
X2
39. In addition, K Q[X]/
+ 39 implies n = [K : Q] = 2, r = 0, and
Math 567 Number Theory I
Homework 4
Name: Zhaoning Yang
October 7, 2013
Problem 1
(Textbook P.107 #4) Determine
irrQ(i)
1+i
2
and irrQ(2)
1+i
2
Solution:
Proof. Let = (1 + i)/ 2. We claim
irrQ(i) () = x2 i and irrQ(2) () = x2
Frist is a root of x2 i and
Math 567 Number Theory I
Homework 6
Name: Zhaoning Yang
October 21, 2013
Problem 1
(Textbook P.189 #1) Let D denote the discriminate of
f (x) = xn + an1 xn1 + + a1 x + a0 Z[x]
Prove that D 0 or 1(mod 4).
Solution:
Proof. Let f (x) = xn +an1 xn1 + +a1 x+a0
Math 567 Number Theory I
Homework 9
Name: Zhaoning Yang
Due: Monday, November 11, 2013
Problem 1
(Textbook P.261 #3) Factor 6 into prime ideals in OQ366 .
Solution: Its clear that 6 = 2 3 , so we only need to gure out how 2, 3 factors in OK where
K = Q( 3
Math 567 Number Theory I
Homework 10
Name: Zhaoning Yang
December 1, 2013
Problem 1
(Textbook P.341 #2) Prove that h(Q( 7) = 1.
Solution:
Proof. Let K = Q( 7), we want to show h(K) = 1. Recall the m function dened on Q( m) for
every m 1(mod 4) square free
Math 567 Number Theory I
Homework 8
Name: Zhaoning Yang
November 3, 2013
Problem 1 (Textbook P.234 #9.13) Let K be an algebraic number eld and OK be its ring of integers.
Let m Z\cfw_0. Prove that there exists only nitely many integral ideals of OK to whi
Math 567 Number Theory I
Homework 1
Name: Zhaoning Yang
September 13, 2013
Problem 1
(Chapter 1, #2) Prove that U (Z + Z) = cfw_1, , 2
Solution:
Proof. It is clear that 1, , 2 U (Z + Z), we only need to prove the reverse inclusion. Suppose
a + b U (Z + Z