Field of Fractions

Abstract Algebra · Axiom Academy

Constructing Frac(D) from an integral domain D: How every integral domain embeds into a field, just like ℤ embeds into ℚ An integral domain D is a commutative ring with no zero divisors. We can add, subtract, and multiply, but we can't always divide. We start with the set of all ordered pairs (a, b) where a, b D and . Think of (a, b) as " a/b ". The field of fractions is the set of equivalence classes [a, b] under this relation. We write this as a/b when the context is clear. We define addition and multiplication on using the familiar rules for fractions: Two fundamental examples show the power of this construction: The field of fractions is characterized by a beautiful universal property:

This is the written version of the interactive lesson above. See the full Abstract Algebra course.