 Solution

ID: past-exam-of-the-algebra-analysis-and-differential-equations-2-course-of-the-university-of-oxford/2010/algebra/3/a/solution

Interpret membership as having a representation with denominator coprime to

30

, equivalently having this property in lowest terms. Clearly

0, 1 \in R

. If

a / b, c / d \in R

with admissible denominators, then

\frac{a}{b} - \frac{c}{d} = \frac{a d - b c}{b d}, \frac{a}{b} \frac{c}{d} = \frac{a c}{b d} .

The product

b d

is still coprime to

30

. These are elements of

R

, proving that

R is a subring of Q

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