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Mathematics 124
IDEALS, PART I
In these notes we discuss some theorems from the theory of ideals, several of which are not
covered in detail in the texts.
Throughout, let d be a squarefree integer, let K = Q( d), and let R be the ring of algebraic
integer
Mathematics 124
HOMEWORK ASSIGNMENT # 9
DUE, Monday, December 15
Collaboration: On the homework sets, collaboration is not only allowed but encouraged.
However, you must write up and understand your own individual homework solutions, and
you may not share
Mathematics 124
IDEALS, PART II
In these notes we discuss some theorems from the theory of ideals, several of which are not
covered in detail in the texts.
Throughout, let d be a squarefree integer, let K = Q( d), and let R be the ring of algebraic
intege
Mathematics 124
HOMEWORK ASSIGNMENT # 8
DUE, Friday, December 5
Collaboration: On the homework sets, collaboration is not only allowed but encouraged.
However, you must write up and understand your own individual homework solutions, and
you may not share
Mathematics 124
HOMEWORK ASSIGNMENT # 6
DUE, Friday, November 14
Collaboration: On the homework sets, collaboration is not only allowed but encouraged.
However, you must write up and understand your own individual homework solutions, and
you may not share
Math 124 Problem Set 1 Solutions
Problem 1: (a) Note that
n
n1
2n = 2n +
i=0
2i = 2n + 2n 1 = 2n+1 1.
i=0
(b) Note that
3
n
i
n1
2
n1
=
i=1
i=1
n1
i + n2 =
+ 2n
i=1
n
i3 + n2 (n 1) + n2 =
i=1
i3 .
i=1
(c) Note that
e
pi = p
d=
d|pe
Pe
i=0
i
= pe(e+1)/2 =
Math 124 Homework 7 Solutions
1a. Let = a + b 1+ 2 p , with a, b Z. We have
2a + b + b p
2a + b b p
N () =
2
2
(2a + b)2 + pb2
4
2
Now, if b = 0, then N () = a = q so b = 0. In particular, b2 1, and we
have N () 1+p .
4
=
1b. Since p is a quadratic residu
Math 124 Homework 6 Solutions
1. For n = 0, the continued fraction is simply 1. For n < 0, the continued
fraction is the same as for n, so we only need to
consider the case n > 0.
The rst term is the greatest integer less than n2 + 1, which is n. The
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Mathematics 124
HOMEWORK ASSIGNMENT # 7
DUE, Friday, November 21
Collaboration: On the homework sets, collaboration is not only allowed but encouraged.
However, you must write up and understand your own individual homework solutions, and
you may not share