The strictly positive integers can be decomposed into a product of prime powers. Likewise, the positive rationals can be decomposed into a product of (possibly negative) prime powers.
Another way to state this is that the underlying multiplicative monoids are free and admit the primes as a basis.
If negative integers or rationals are allowed, then we no longer get a free monoid. But, we can still get a suitable prime decomposition if we create an additional "basis" element, representing -1, that takes exponents in $\Bbb Z/ 2 \Bbb Z$.
I'm wondering if a similar sort of decomposition exists for any of the more interesting structures thrown around in number theory, namely
- The profinite integers, and/or its group of units
- The ring of adeles, and/or its group of units
- The field of p-adic numbers for some p
I'm particularly interested in the case where the "basis" is countable, like the primes themselves.
Now, since these are all uncountable, it's clear that a decomposition into a direct sum of countable groups won't work. But perhaps we can allow infinitely many non-zero exponents, or allow the exponents to take values in some interesting group. Does anything like this work?
The supernatural numbers seem related here, but I can't tell exactly what the relationship is, or if they embed into one of the groups mentioned (or vice versa).