Prove that ${\mathbb{Q}}/{\mathbb{Z}}$ is isomorphic to $\bigoplus_p \mathbb{Z}[p^{\infty}]$

Question

Prove that ${\mathbb{Q}}/{\mathbb{Z}}$ is isomorphic to $\bigoplus_p \mathbb{Z}[p^{\infty}]$

I know we will use $A_p$'s for solution. But i dont know how to do the isomorphism between ${\mathbb{Q}}/{\mathbb{Z}}$ and $\bigoplus_p \mathbb{Z}[p^{\infty}]$.

Answer 1

Hint: the exponential map

$$ \mathbf{exp} \colon t \in \Bbb R \mapsto e^{i2\pi t} \in S^1 $$

is a surjective group homomorphism with kernel $\Bbb Z$, hence

$$ S^1 = \Bbb R/\Bbb Z. $$

Via this correspondence, elements of $\Bbb Q / \Bbb Z$ are complex elements of absolute value $1$ of the form $e^{i2\pi p/q}$ with $p/q \in \Bbb Q$. Can you think of a way to embed $\Bbb Z[p^\infty]$ in $S^1$?

Note that elements of the form $e^{i2\pi r}$ with rational $r$ are roots of unity, and conversely roots of unity also have this expression. Maybe you can 'classify' roots of unity in terms of primes?

Hint': Note that $\Bbb Z[p^\infty]$ is the $p$-torsion of $\Bbb Q/ \Bbb Z$, then use primary decomposition. This may be easier to do with the aforemention description (or not).

Hint'': show that if $\xi_n$ and $\xi_m$ are primitive roots of unity of coprime orders $n$ and $m$, then $\xi_n\xi_m$ is primitive of order $nm$. Deduce that any root of unity is a finite product of powers of primitive roots of unity of order $p^r$ for $p$ prime and $r \geq 0$. Use that to construct a map $\bigoplus_p \Bbb Z[p^\infty] \to S^1$ whose image is (the isomorphic copy of) $\Bbb Q / \Bbb Z$, prove that it is also injective.

Answer 2

Just wanted to provide some interesting context.

You may be familiar with partial fraction decomposition for rational functions. Any rational function $f/g$ is expressible as $f(T)/g(T)=p(T)+\sum a_k(T)/\pi_k(T)^{e_k}$ where $p(T)$ is a polynomial, $g(T)=\prod_k \pi_k(T)^{e_k}$ is a factorization into irreducible polynomials, and $a_k(T)$s are polynomials with $\deg a_k< e_k\deg \pi_k$ (so none of the summands can be partial-fractioned more in this fashion). One could choose to reduce $a(T)/\pi(T)^e$ to the form $\sum_{1<j\le e} a_j/\pi(T)^j$ for some scalar coefficients $a_j$.

Ultimately, if $\mathbb{F}$ is a field, this can be phrased as $\mathbb{F}(T)/\mathbb{F}[T]=\bigoplus \mathbb{F}[\pi(T)^{-1}]/\mathbb{F}[T]$.

Similarly, $\mathbb{Q}/\mathbb{Z}=\bigoplus \mathbb{Z}[1/p]/\mathbb{Z}$ can be phrased as follows: every rational number $f/g$ is uniquely expressible as $f/g=n+\sum a_k/p_k^{e_k}$ for some integer $n$, factorization $g=\prod p_k^{e_k}$, and coefficients $0<a_k<p_k^{e_k}$, and we can rewrite $b/p^e$ as $\sum_{1\le j\le e} b_j/p^j$ for coefficients $0<b_j<p$ if desired.

You can solve for the $a_k$s by multiplying $f/g$ by $p_k^{e_k}$ and reducing mod $p_k^{e_k}$.