Quotient Field

• Every Integral Domain is contained in a field — the field of quotients of the integral domain

• (Fraleigh 21.2) The relation is an equivalence Relation. Where and

  • Intuition: A more suggestive way to say this is that
    is defined such that is the numerator and is the denominator.

• (Fraleigh 21.3) The following give well-defined operations of addition and multiplication on where

  We must show that using the relation in (Fraleigh 21.2) and , then

  • Intuition: A more suggestive way to view this is as the product of two fractions.

• (Fraleigh 21.4) The map given by is an isomorphism of with a subring of .

• (Fraleigh 21.5) Any integral domain can be enlarged to a field such that every element of can be expressed as a quotient of two elements of . We call this field the field of quotients of . We denote this as

  • Intuition: A suggestive example: . In other words, the set of integers (integral domain) is contained in the set of possible quotients (field of quotients). In particular .
  • The field of quotients is the minimal field containing . This follows since a field containing must satisfy the property of multiplicative inverses

• (Fraleigh 21.6) Let be a field of quotients of and be any field containing . Then there exists a map that gives an isomorphism of with a subfield of such that

  In fact such a map can be defined using the quotient of over (denoted )

  This division operator is defined such that

  Where

• (Fraleigh 21.8) Every field containing an integral domain contains a field of quotients of .

• (Fraleigh 21.9) Any two fields of quotients of an integral domain are isomorphic

• (Fraleigh 25.13) Let be an ordered integral domain with as the set of positive elements. The set

  is well defined and gives an order of on that induces the given order on . Furthermore, is the only subset of with this property.

Localization of a Ring §

• In general a Commutative Ring that is not an integral domain can be enlarged to a field.

  Let be a nonzero commutative ring and be a non-empty subset of closed under multiplication and containing neither nor divisors of .

  Using the same enlargement for the quotient field of an integral domain, we can enlarge to a partial ring of quotients .

  • (Fraleigh e21.12a) has unity even if does not.
  • (Fraleigh e21.12b) In , every nonzero element of is a unit.

Links §

• Fraleigh