Dartmouth College
Mathematics 81/111 Homework 3 - solutions
1. A rst approximation to a theorem of Bezout.
(a) Let A be a UFD with eld of fractions F . Let f, g A[x]. Show that f, g are
relatively prime in A[x] if and only if f, g are relatively prime in
Dartmouth College
Mathematics 81/111 Homework 2
1. Prime and maximal ideals in non-commutative rings. We start with two basic denitions:
Denition: A ring A is simple if its only (2-sided) ideals are cfw_0 and A. We
know that if k is a eld, that k is a si
Dartmouth College
Mathematics 81/111 Homework 1 Solutions
Some basic denitions concerning algebraic sets
Let k be a eld, and let k [x1 , . . . , xn ] the polynomial ring in n variables with coecients
in k . For f k [x1 , . . . , xn ] and P = (a1 , . . . ,
Dartmouth College
Mathematics 81/111 Homework 4
1. Let F be a eld of characteristic 0, and let m and n be distinct integers with
n F , and mn F .
/
/
(a) Show that [F ( m, n) : F ] = 4.
m F,
/
(b) Show by example (with m/n (Q 2 ) that the above statement
Dartmouth College
Mathematics 81/111 Homework 3
1. A rst approximation to a theorem of Bezout.
(a) Let A be a UFD with eld of fractions F . Let f, g A[x]. Show that f, g are
relatively prime in A[x] if and only if f, g are relatively prime in F [x] and (t