Let D be an integral domain and let F be the field of...

Let $D$ be an integral domain and let $F$ be the field of quotients of $D$. Show that if $E$ is any field that contains $D$, then $E$ contains a subfield that is ring-isomorphic to $F$. (Thus, the field of quotients of an integral domain $D$ is the smallest field containing $D .$ )

Key Concepts

Universal Property

The universal property of the field of quotients states that for any integral domain, there exists a unique (up to isomorphism) smallest field that contains it, and every homomorphism from the integral domain to any field factors uniquely through its field of quotients. This property is essential for demonstrating the minimal and canonical nature of the field of quotients among all fields that contain the integral domain.

Ring Isomorphism

Ring isomorphism is a bijective ring homomorphism that preserves both the addition and multiplication operations. Proving that two structures are ring-isomorphic means they share the same algebraic properties, even if their elements or representations differ. In the context of the field of quotients, a ring isomorphism ensures that any field containing the integral domain has a copy of the field of fractions within it.

Field of Quotients

The field of quotients (or field of fractions) of an integral domain is the smallest field in which the domain can be embedded. It is constructed by considering fractions of elements from the integral domain, analogous to constructing the rational numbers from the integers. This construction satisfies a universal property, making it unique up to isomorphism for any given integral domain.

Integral Domain

An integral domain is a commutative ring with unity that contains no zero divisors. This property ensures that the cancellation law holds, making it possible to construct fractions in a consistent manner. Integral domains serve as the basic building blocks in ring theory for creating larger structures like fields through processes such as forming the field of quotients.

Subfield

A subfield of a field is a subset that is itself a field under the same operations as the larger field. Demonstrating that a given field contains a subfield isomorphic to the field of quotients shows that the construction of the field of quotients is intrinsic and minimal, serving as the core field structure associated with the integral domain.

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