What does $p$-integral mean? I'm currently studying Washington's Introduction to Cyclotomic fields and in Theorem 5.10 I came across the term $p$-integral. What does this mean? 
To give a bit of context: Let $n$ be even and positive. By the von Staudt - Clausen Theorem we have $$B_n + \sum_{(p - 1) \mid n} \frac{1}{p} \in \mathbf{Z}$$ where $B_n$ is the $n$-th Bernoulli-number. Now the author states that "consequently $p B_n$ is $p$-integral for all $n$ and all $p$".
Thanks!
 A: In this context, it is a property of rational numbers.  A rational number is $p$-integral if, when written in lowest terms, there are no factors of $p$ in the denominator.  
This definition can be extended to other rings of integers, and is best understood through the concept of valuations.  If $K$ is an algebraic number field and $\mathfrak{P}$ is a prime ideal in its ring of integers, then an element of $K$ is $\mathfrak{P}$-integral if its $\mathfrak{P}$-valuation is nonnegative.
Some nice exercises:
Show that a sum or product of $p$-integral rational numbers is again $p$-integral.
Show that if a rational number is $p$-integral for all primes $p$, then it is an integer. 
A: A rational number $r$ is called $p$-integral if $ord_p(r)\ge 0$. For example $\frac{p^k}{k+1}$ is $p$-integral for any $k\ge 1$. It is a Lemma that  $pB_n$ is $p$-integral for every prime and every integer $n$. You don't need the Staudt-Clausen theorem for it, usually it is the other way around, that you use the Lemma for the proof of Staudt-Clausen.
