MATH 431 PART 1: FIELDS OF QUOTIENTS
Over-arching goal of field theory: We want to understand where solutions to polynomials live!
Humans have been thinking about polynomials for over 2,000 years. Mathematicians have been fascinated by roots
of polynomials for many of those. Our goal for this course is to understand the field in which a polynomial’s roots
lie. We will start by thinking about integral domains, because rings like Z[x] are of interest to many different kinds
mathematicians but the roots of these polynomials rarely live in Z themselves.
We really like integral domains because there are no zero divisors, and so our normal laws of algebra hold. We also
really like fields (multiplicative inverses are good) but not all integral domains are fields. Take Z for example; what is
the smallest field that contains Z and all of its multiplicative inverses? Can we construct it?
1. Fields of Quotients of Integral Domains
Example 1 (Motivating example). The field of quotients of an integral domain D is the smallest field which contains
D in some sense. The field of quotients of Z is Q, the field constructed of quotients of elements of Z. The field of
quotients of Z can’t be R because there is a smaller field (Q ( R) that already contains all quotients of integers.
Theorem 1. Every integral domain D can be enlarged to (or embedded in) a field F such that every element of F can
be expressed as a quotient of two elements of D. We call F the field of quotients of D.
Proof. Let D be an integral domain. We will prove this theorem by constructing a field F which is its field of quotients.
(Step 1.) Consider D × D = {(a, b) | a, b ∈ D}.
Define two elements (a, b) and (c, d) of S to be equivalent if
1
2
MATH 431 PART 1: FIELDS OF QUOTIENTS
(Step 2.) Let F be the set of equivalence classes of elements of S. (Denote the equivalence class of (a, b) ∈ S by
[(a, b)].) Define addition and multiplication on F as
[(a, b)] + [(c, d)] =
[(a, b)][(c, d)] =
Fact: These operations are well-defined; in other words, if we pick different representatives of the equivalence
classes, these operations will produce representatives of the same class. (The proof of this is in the book on
page 192.)
(Step 3.) Under these operations, F is a field.
Notice that F really does contain the elements of D, since for a ∈ D, [(a, 1)] ∈ F . (You can show the map φ : D → F
with φ(a) = [(a, 1)] is a ring homomorphism, and an isomorphism with a subring of F .) Then we have proved (by
construction) the theorem.
Theorem 2. Let F be a field of quotients for D and let L be any field containing D. Then there exists a map
ψ : F → L that gives an isomorphism between F and a subfield of L such that ψ(a) = a for all a ∈ D.
Corollary 3. Every field L containing an integral domain D contains a field of quotients of D.
Corollary 4. If F1 and F2 are both fields of quotients for D, then F1 and F2 are isomorphic as fields.