Undergraduate Texts in Mathematics
Editors
F. W. Gehring
P. R. Halmos
Advisory Board
C. DePrima
I. Herstein
J. Kiefer
W. LeVeque
Tom M. Apostol
Introduction
to Analytic
Number Theory
Springer-Verlag
New York Heidelberg Berlin
1976
Tom M. Apostol
Professor

Selected Solutions to Artins Algebra, Second Ed.
Takumi Murayama
July 22, 2014
These solutions are the result of taking MAT323 Algebra in the Spring of 2012,
and also TA-ing for MAT346 Algebra II in the Spring of 2014, both at Princeton
University. This i

Math 122, Solution Set No. 13
1
10.3.4
Let f (2) (x) Z[x]. Since x - 2 and 2 - x, we have f = 2xg(x) and so f (2x). Conversely, if
f (2x) Z[x] we have f = 2xg(x) f (2) (x). (2) (x) = (2x) Z[x].
2
10.3.6
Pn
Let I := cfw_ 0 ak xk Z[x]|2k+1 |ak . Note 2 I, b

Math 113 Homework 5
Solutions
due July 24, 2011
1. How many abelian groups are there of order 100? of order 16, 900?
Solution: 100 = 22 52 , so by the structure theorem for finite abelian
groups, there are 4 (namely, (Z/2Z)2 (Z/5Z)2 , (Z/2Z)2 Z/25Z, Z/4Z