Let R be a ring. We say that R is a domain if for all a, b R we have
ab = 0
a = 0 or b = 0,
that is, if the ring has no zerodivisors.
1. Prime Ideals. Given an ideal I R in a general ring R we say that I is
Problems on Integers.
1. Z[ 1] is Euclidean. Historically, the rst Euclidean domain considered (by Gauss)
beyond Z and Q[x] was the ring of Gaussian integers:
Z[ 1] := cfw_a + b 1 : a, b Z.
(a) We can think o
Problems on Rings
1. Chinese Remainder Theorem. Given two ideals I, J R we dene their product:
IJ := cfw_ab : a I, b J .
This is the smallest ideal containing all the elements ab for a I and b J.
(a) Prove th
1. (Finite Implies Algebraic) Consider a eld extension L K. Recall that we say L
is algebraic over K if there exists nonzero f (x) K[x] such that f () = 0. We say that the
eld extension K L is algebraic if ev
Problems on Integers
1. The Division Algorithm. Consider integers a, b Z with b = 0.
(a) Prove that there exist integers q, r Z such that a = qb + r and 0 r < |b|. [Hint: Let
S be the set of integers of the f