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what does it means that a place divide a prime on an algebraic number field?

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A place is a valuation (or an equivalence class of) $v:K^\times\rightarrow{\Bbb Z}$. The place $v$ divides the rational prime $p$ when $v(p)>0$, or equivalently when $v$ extends the $p$-adic valuation on $\Bbb Q$.

Since (non-archimedean) places in a number field $K$ correspond to prime non-zero ideals in the ring of integers ${\cal O}_K$, another formulation is that the valuation $v$ corresponding to the prime ideal ${\cal P}_v\subset{\cal O}_K$ divides $p$ when ${\cal P}_v$ appears in the primary decomposition of the ideal $p{\cal O}_K$.

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I think you mean $v(p) > 0$. –  Pete L. Clark Feb 10 '11 at 17:16
    
Of course, Pete. I'll edit –  Andrea Mori Feb 10 '11 at 20:01

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