有理数的质因数分解

We already know that all positive integers (other than the unit $1$) have a unique prime factorization, this is the Fundamental Theorem of Arithmetic (proven here).

Here, we left out $1$ because it has no prime divisors. Another way to look at it, however, is as a number whose prime factorization contains only $0$ exponents. Naturally, $1=2^0=2^0\cdot 3^0\cdot 5^0$ etc., but if we allow zero exponents, the factorization ceases to be unique.

A way to implement $0$ exponents in the prime factorization of $n$ without losing unicity would be just to imagine all the prime numbers in the product, and simply give a $0$ exponent to those not dividing $n$. The exponent of an arbitrary prime $p$ in this product is denoted by $v_p(n)$ (called the $p$-adic valuation of $n$). Thus, we may write

$$n=2^{v_2(n)}\cdot 3^{v_3(n)}\cdot 5^{v_5(n)}\cdot \ldots=\prod_{p\text{ prime}} p^{v_p(n)}.$$

In fact, we may identify each positive integer with the sequence $(v_2(n),v_3(n),v_5(n),...)$. We may find a prime factorization for all the positive rational numbers in a similar fashion. Since we must allow reciprocation, we should allow negative integer exponents in this sequence as well.

Having done that, we may quickly extend unique factorization. Essentially, a prime factorization of a rational number $q$ means a sequence of integer exponents $(v_2(q),v_3(q),v_5(q),...)$ - only finitely many of whom are nonzero -, which satisfies

$$q=\prod_{p\text{ prime}} p^{v_p(q)}.$$

We claim that every rational number has a unique prime factorization. Clearly, such a factorization exists - just write $q$ as a fraction $\frac ab$ and take the primes in $b$ with negative exponents, i.e. let $v_p(\frac ab)=v_p(a)-v_p(b)$. Moreover, this factorization is unique because if we had two different factorizations, then we could multiply $q$ by an appropriate positive integer to obtain two different factorizations of a positive integer, contradicting the Fundamental Theorem of Arithmetic.

(Maybe this last sentence was a bit unclear, so let's take an example - say if $(-1,2,3,0,0,0,...)$ and $(2,-3,4,0,0,0,...)$ yields the prime factorizations of the same number, then multiplying this number by $2^1\cdot 3^3$, we see that $(0,5,3,0,0,0,...)$ and $(3,0,4,0,0,0,...)$ give two different prime factorizations of a positive integer, contradiction.)

The concept of $p$-adic valuation is useful mainly as a notation, and it also gives deeper insight into prime factorization. Try to prove some of its properties in the following problems.