18.1: Fields of Fractions
Every field is also an integral domain; however, there are many integral domains that are not fields. For example, the integers \({\mathbb Z}\) form an integral domain but not a field. A question that naturally arises is how we might associate an integral domain with a field. There is a natural way to construct the rationals \({\mathbb Q}\) from the integers: the rationals can be represented as formal quotients of two integers. The rational numbers are certainly a field. In fact, it can be shown that the rationals are the smallest field that contains the integers. Given an integral domain \(D\text{,}\) our question now becomes how to construct a smallest field \(F\) containing \(D\text{.}\) We will do this in the same way as we constructed the rationals from the integers.
An element \(p/q \in {\mathbb Q}\) is the quotient of two integers \(p\) and \(q\text{;}\) however, different pairs of integers can represent the same rational number. For instance, \(1/2 = 2/4 = 3/6\text{.}\) We know that
\[ \frac{a}{b} = \frac{c}{d} \nonumber \]
if and only if \(ad = bc\text{.}\) A more formal way of considering this problem is to examine fractions in terms of equivalence relations. We can think of elements in \({\mathbb Q}\) as ordered pairs in \({\mathbb Z} \times {\mathbb Z}\text{.}\) A quotient \(p/q\) can be written as \((p, q)\text{.}\) For instance, \((3, 7)\) would represent the fraction \(3/7\text{.}\) However, there are problems if we consider all possible pairs in \({\mathbb Z} \times {\mathbb Z}\text{.}\) There is no fraction \(5/0\) corresponding to the pair \((5,0)\text{.}\) Also, the pairs \((3,6)\) and \((2,4)\) both represent the fraction \(1/2\text{.}\) The first problem is easily solved if we require the second coordinate to be nonzero. The second problem is solved by considering two pairs \((a, b)\) and \((c, d)\) to be equivalent if \(ad = bc\text{.}\)
If we use the approach of ordered pairs instead of fractions, then we can study integral domains in general. Let \(D\) be any integral domain and let
\[ S = \{ (a, b) : a, b \in D \text{ and } b \neq 0 \}\text{.} \nonumber \]
Define a relation on \(S\) by \((a, b) \sim (c, d)\) if \(ad = bc\text{.}\)
Lemma \(18.1\)
The relation \(\sim\) between elements of \(S\) is an equivalence relation.
- Proof
-
Since \(D\) is commutative, \(ab = ba\text{;}\) hence, \(\sim\) is reflexive on \(D\text{.}\) Now suppose that \((a,b) \sim (c,d)\text{.}\) Then \(ad=bc\) or \(cb = da\text{.}\) Therefore, \((c,d) \sim (a, b)\) and the relation is symmetric. Finally, to show that the relation is transitive, let \((a, b) \sim (c, d)\) and \((c, d) \sim (e,f)\text{.}\) In this case \(ad = bc\) and \(cf = de\text{.}\) Multiplying both sides of \(ad = bc\) by \(f\) yields
\[ a f d = a d f = b c f = b d e = bed\text{.} \nonumber \]
Since \(D\) is an integral domain, we can deduce that \(af = be\) or \((a,b ) \sim (e, f)\text{.}\)
We will denote the set of equivalence classes on \(S\) by \(F_D\text{.}\) We now need to define the operations of addition and multiplication on \(F_D\text{.}\) Recall how fractions are added and multiplied in \({\mathbb Q}\text{:}\)
\begin{align*} \frac{a}{b} + \frac{c}{d} & = \frac{ad + b c}{b d};\\ \frac{a}{b} \cdot \frac{c}{d} & = \frac{ac}{b d}\text{.} \end{align*}
It seems reasonable to define the operations of addition and multiplication on \(F_D\) in a similar manner. If we denote the equivalence class of \((a, b) \in S\) by \([a, b]\text{,}\) then we are led to define the operations of addition and multiplication on \(F_D\) by
\[ [a, b] + [c, d] = [ad + b c,b d] \nonumber \]
and
\[ [a, b] \cdot [c, d] = [ac, b d]\text{,} \nonumber \]
respectively. The next lemma demonstrates that these operations are independent of the choice of representatives from each equivalence class.
Lemma \(18.2\)
The operations of addition and multiplication on \(F_D\) are well-defined.
- Proof
-
We will prove that the operation of addition is well-defined. The proof that multiplication is well-defined is left as an exercise. Let \([a_1, b_1] = [a_2, b_2]\) and \([c_1, d_1] =[ c_2, d_2]\text{.}\) We must show that
\[ [a_1 d_1 + b_1 c_1,b_1 d_1] = [a_2 d_2 + b_2 c_2,b_2 d_2] \nonumber \]
or, equivalently, that
\[ (a_1 d_1 + b_1 c_1)( b_2 d_2) = (b_1 d_1) (a_2 d_2 + b_2 c_2)\text{.} \nonumber \]
Since \([a_1, b_1] = [a_2, b_2]\) and \([c_1, d_1] =[ c_2, d_2]\text{,}\) we know that \(a_1 b_2 = b_1 a_2\) and \(c_1 d_2 = d_1 c_2\text{.}\) Therefore,
\begin{align*} (a_1 d_1 + b_1 c_1)( b_2 d_2) & = a_1 d_1 b_2 d_2 + b_1 c_1 b_2 d_2\\ & = a_1 b_2 d_1 d_2 + b_1 b_2 c_1 d_2\\ & = b_1 a_2 d_1 d_2 + b_1 b_2 d_1 c_2\\ & = (b_1 d_1) (a_2 d_2 + b_2 c_2)\text{.} \end{align*}
Lemma \(18.3\)
The set of equivalence classes of \(S\text{,}\) \(F_D\text{,}\) under the equivalence relation \(\sim\text{,}\) together with the operations of addition and multiplication defined by
\begin{align*} [a, b] + [c, d] & = [ad + b c, b d]\\ [ a, b] \cdot [c, d] & = [ac, b d]\text{,} \end{align*}
is a field.
- Proof
-
The additive and multiplicative identities are \([0,1]\) and \([1,1]\text{,}\) respectively. To show that \([0,1]\) is the additive identity, observe that
\[ [a, b] + [0, 1] = [ a 1 + b 0, b 1] = [a,b]\text{.} \nonumber \]
It is easy to show that \([1, 1]\) is the multiplicative identity. Let \([a, b] \in F_D\) such that \(a \neq 0\text{.}\) Then \([b, a]\) is also in \(F_D\) and \([a,b] \cdot [b, a] = [1,1]\text{;}\) hence, \([b, a]\) is the multiplicative inverse for \([a, b]\text{.}\) Similarly, \([-a,b]\) is the additive inverse of \([a, b]\text{.}\) We leave as exercises the verification of the associative and commutative properties of multiplication in \(F_D\text{.}\) We also leave it to the reader to show that \(F_D\) is an abelian group under addition.
It remains to show that the distributive property holds in \(F_D\text{;}\) however,
\begin{align*} [a, b] [e, f] + [c, d][ e, f ] & = [a e, b f ] + [c e, d f]\\ & = [a e d f + b f c e, b d f^2 ]\\ & = [a e d + b c e, b d f ]\\ & = [a d e + b c e, b d f ]\\ & = ( [a, b] + [c, d] ) [ e, f ] \end{align*}
and the lemma is proved.
The field \(F_D\) in Lemma 18.3 is called the field of fractions or field of quotients of the integral domain \(D\text{.}\)
Theorem \(18.4\)
Let \(D\) be an integral domain. Then \(D\) can be embedded in a field of fractions \(F_D\text{,}\) where any element in \(F_D\) can be expressed as the quotient of two elements in \(D\text{.}\) Furthermore, the field of fractions \(F_D\) is unique in the sense that if \(E\) is any field containing \(D\text{,}\) then there exists a map \(\psi : F_D \rightarrow E\) giving an isomorphism with a subfield of \(E\) such that \(\psi(a) = a\) for all elements \(a \in D\text{,}\) where we identify \(a\) with its image in \(F_D\text{.}\)
- Proof
-
We will first demonstrate that \(D\) can be embedded in the field \(F_D\text{.}\) Define a map \(\phi : D \rightarrow F_D\) by \(\phi(a) = [a, 1]\text{.}\) Then for \(a\) and \(b\) in \(D\text{,}\)
\[ \phi( a + b ) = [a+b, 1] = [a, 1] + [b, 1] = \phi(a ) + \phi(b) \nonumber \]
and
\[ \phi( a b ) = [a b, 1] = [a, 1] [b, 1] = \phi(a ) \phi(b); \nonumber \]
hence, \(\phi\) is a homomorphism. To show that \(\phi\) is one-to-one, suppose that \(\phi(a) = \phi( b)\text{.}\) Then \([a, 1] = [b, 1]\text{,}\) or \(a = a1 = 1b = b\text{.}\) Finally, any element of \(F_D\) can be expressed as the quotient of two elements in \(D\text{,}\) since
\[ \phi(a) [\phi(b)]^{-1} = [a, 1] [b, 1]^{-1} = [a, 1] \cdot [1, b] = [a, b]\text{.} \nonumber \]
Now let \(E\) be a field containing \(D\) and define a map \(\psi :F_D \rightarrow E\) by \(\psi([a, b]) = a b^{-1}\text{.}\) To show that \(\psi\) is well-defined, let \([a_1, b_1] = [a_2, b_2]\text{.}\) Then \(a_1 b_2 = b_1 a_2\text{.}\) Therefore, \(a_1 b_1^{-1} = a_2 b_2^{-1}\) and \(\psi( [a_1, b_1]) = \psi( [a_2, b_2])\text{.}\)
If \([a, b ]\) and \([c, d]\) are in \(F_D\text{,}\) then
\begin{align*} \psi( [a, b] + [c, d] ) & = \psi( [ad + b c, b d ] )\\ & = (ad + b c)(b d)^{-1}\\ & = a b^{-1} + c d^{-1}\\ & = \psi( [a, b] ) + \psi( [c, d] ) \end{align*}
and
\begin{align*} \psi( [a, b] \cdot [c, d] ) & = \psi( [ac, b d ] )\\ & = (ac)(b d)^{-1}\\ & = a b^{-1} c d^{-1}\\ & = \psi( [a, b] ) \psi( [c, d] )\text{.} \end{align*}
Therefore, \(\psi\) is a homomorphism.
To complete the proof of the theorem, we need to show that \(\psi\) is one-to-one. Suppose that \(\psi( [a, b] ) = ab^{-1} = 0\text{.}\) Then \(a = 0b = 0\) and \([a, b] = [0, b]\text{.}\) Therefore, the kernel of \(\psi\) is the zero element \([ 0, b]\) in \(F_D\text{,}\) and \(\psi\) is injective.
Example \(18.5\)
Since \({\mathbb Q}\) is a field, \({\mathbb Q}[x]\) is an integral domain. The field of fractions of \({\mathbb Q}[x]\) is
Solution
the set of all rational expressions \(p(x)/q(x)\text{,}\) where \(p(x)\) and \(q(x)\) are polynomials over the rationals and \(q(x)\) is not the zero polynomial. We will denote this field by \({\mathbb Q}(x)\text{.}\)
We will leave the proofs of the following corollaries of Theorem \(18.4\) as exercises.
Corollary \(18.6\)
Let \(F\) be a field of characteristic zero. Then \(F\) contains a subfield isomorphic to \({\mathbb Q}\text{.}\
Corollary \(18.7\)
Let \(F\) be a field of characteristic \(p\text{.}\) Then \(F\) contains a subfield isomorphic to \({\mathbb Z}_p\text{.}\)