Abstract Algebra: Theory and Applications · Problem 18.4.7

Lattices and Boolean Algebras — Thomas Judson

Mathematics — Abstract Algebra · field-of-fractions-polynomial-ring

Problem

Let

F

be a field. ((a)) Prove that the field of fractions of

F[x]

, denoted by

F(x)

, is isomorphic to the set of all rational expressions

p(x)/q(x)

, where

q(x)

is not the zero polynomial. ((b)) Let

p(x_1, \ldots, x_n)

and

q(x_1, \ldots, x_n)

be polynomials in

F[x_1, \ldots, x_n]

. Show that the set of all rational expressions

p(x_1, \ldots, x_n)/q(x_1, \ldots, x_n)

is isomorphic to the field of fractions of

F[x_1, \ldots, x_n]

. We denote the field of fractions of

F[x_1, \ldots, x_n]

F(x_1, \ldots, x_n)