Or is this algebra

Number Theory Level 3

Consider $\mathbb{Q}$ as a left $\mathbb{Z}$ -module (Where addition and scalar multiplication are as usual when considering $\mathbb{Q}$ as a ring and $\mathbb{Z}$ as a subring of $\mathbb{Q}$ ). Does $\{\frac{1}{p}|p\text{ prime}\}$ span $\mathbb{Q}$ ?

No Yes

1 solution

Jonathan Dunay
Dec 10, 2017

No.

Any linear combination of reciprocals of odd primes will have the form $\frac{a}{b}$ for some $a,b \in \mathbb{Z}$ where $b \neq 0$ is odd. So any linear combination of reciprocals of primes will have the form $\frac{c}{2} + \frac{a}{b} = \frac{bc+2a}{2b}$ which cannot be equal to $\frac{1}{4}$ . So $\frac{1}{4}$ is not in the span of $\{\frac{1}{p}|p\text{ prime}\}$

Or is this algebra

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