Exercise · ZW-FIELD-OF-FRACTIONS-008 · ★★☆ Intermediate

Which property characterizes \(\operatorname{Frac}(D)\)?

Choose one option.

Show answer
  • B. It is the smallest field containing \(D\): every field containing \(D\) contains a copy of it

Explanation

A field containing \(D\) must contain the inverses of its nonzero elements and hence all quotients \(ab^{-1}\), which form a copy of \(\operatorname{Frac}(D)\). There is no largest field, it is not always infinite (if \(D\) is a finite field it equals \(D\)), and in general it is not isomorphic to \(D\).

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Review the concept: Field of fractions

The field of fractions \(\operatorname{Frac}(D)\) of an integral domain \(D\) is the field made of the fractions \(a/b\), with \(a,b\in D\) and \(b\neq0\), where \(a/b=c/d\) when \(ad=bc\).

\[\frac ab=\frac cd\iff ad=bc,\qquad \frac ab+\frac cd=\frac{ad+bc}{bd},\qquad \frac ab\cdot\frac cd=\frac{ac}{bd}\]

More practice problems: field of fractions

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