Abstract Algebra: Theory and Applications · Problem 18.4.12

Lattices and Boolean Algebras — Thomas Judson

Mathematics — Abstract Algebra · greatest-common-divisor-pid

Problem

Let

D

be a UFD. An element

d \in D

is a greatest common divisor of

a

and

b

D

d \mid a

and

d \mid b

and

d

is divisible by any other element dividing both

a

and

b

. ((a)) If

D

is a PID and

a

and

b

are both nonzero elements of

D

, prove there exists a unique greatest common divisor of

a

and

b

up to associates. That is, if

d

and

d'

are both greatest common divisors of

a

and

b

, then

d

and

d'

are associates. We write

\gcd(a,b)

for the greatest common divisor of

a

and

b

. ((b)) Let

D

be a PID and

a

and

b

be nonzero elements of

D

. Prove that there exist elements

s

and

t

D

such that

\gcd(a,b) = as + bt