Exercise · ZW-FIELD-OF-FRACTIONS-010 · ★★☆ Intermediate
In \(\operatorname{Frac}(\mathbb{Q}[x])\), what is the multiplicative inverse of \(x+1\)?
Choose one option.
Show answer
- A. \(\dfrac{1}{x+1}\)
Explanation
We have \((x+1)\cdot\tfrac1{x+1}=1\). That inverse is not a polynomial, which is why \(x+1\) is not a unit of \(\mathbb{Q}[x]\), but it is a unit in its field of fractions, where every nonzero element is invertible.
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
- ZW-FIELD-OF-FRACTIONS-001 Which fraction equals tfrac69 in Frac(ℤ)?
- ZW-FIELD-OF-FRACTIONS-002 Which field is Frac(ℤ), the field of fractions of the integers?
- ZW-FIELD-OF-FRACTIONS-003 In Frac(ℤ), what is tfrac12+tfrac13?
- ZW-FIELD-OF-FRACTIONS-004 In Frac(ℤ[x]), what does (x^2-1)/(x-1) equal?
- ZW-FIELD-OF-FRACTIONS-005 For which of these rings can a field of fractions NOT be built?
- ZW-FIELD-OF-FRACTIONS-006 Which of these numbers belongs to Frac(ℤ[i])?
- ZW-FIELD-OF-FRACTIONS-007 If D is already a field, what is Frac(D)?
- ZW-FIELD-OF-FRACTIONS-008 Which property characterizes Frac(D)?