A global field $K$ is defined as a finite extension of $\mathbb Q$ or $\mathbb F_p(t)$. Let's work here with a number field. We will say that two absolute values $|\cdot|_1$, $|\cdot|_2$ of $K$ are related if there is a real $t>0$ such that $|a|_1=|a|_2^t$ for all $a\in K$. A place of $K$ is a class of equivalence of this relation, and the place is called finite if it is non-archimedian (number field case only). We denote the completion of $K$ by a place $\upsilon$ as $K_{\upsilon}$. The ring of integers $\mathfrak{o}_K$ of $K$ is $$\mathfrak o_K=\bigcap_{\upsilon \mbox{ finite}}\{x\in K: |x|_{\upsilon}\leq 1\}.$$
Finally We will say that a prime ideal of $\mathfrak o_K$ is a prime of $K$.
We know that the ring of integers of $K$ is a Dedekind Domain, and that $K$ is it's quotient field. For an element $a\in K$ we have an unique factorization of the fractional ideal $a\mathfrak o_K$ as a product of prime ideals, say $$a\mathfrak o_K=\mathfrak p_{1}^{\alpha_1}\cdots\mathfrak p_{n}^{\alpha_n},$$ with $\alpha_i\in\mathbb Z$. In this way, for every prime $\mathfrak p$ of $K$, we can define an absolute value $|\cdot|_{\mathfrak p}$ by $$|a|_{\mathfrak p}=c^{-ord_{\mathfrak p}(a)},$$ where $c>1$ and $ord_{\mathfrak p}(a)$ is the order of $\mathfrak p$ in the factorization of the fractional ideal $a\mathfrak o_K$. My question is:
Is there a bijective correspondence between the primes of $K$ and it's non-arquimedian places, such as happens with $\mathbb Q$ and $\mathbb Z$?
Any reference is appreciated. I'm working with Cassels's Algebraic number theory but no such relation is mentioned.
Yes, the result you're looking for is Ostrowski's Theorem for number fields. If $ K $ is a number field then upto equivalence, all its valuations are given by a prime ideal in $ O_K $ as you have described (these are non-Archimedian) and by the embeddings of $ K $ into $ \mathbb{R} $ or $ \mathbb{C} $ (these are Archimedian) i.e, if $ \sigma_1, \sigma_2, \ldots, \sigma_{r_1} $ are the real embeddings of $ K $ and $ \tau_1, \bar{\tau_1}, \ldots, \tau_{r_2}, \bar{\tau_{r_2}} $ are the complex embeddings with $ r_1 + 2r_2 = [K:Q] $, then these define absolute values on $ K $ by $ |a|_{\sigma_i} = |\sigma_i(a)| , |a|_{\tau_j} = |\tau_j(a)| $ where the absolute value on the RHS is the usual absolute value on $ \mathbb{R} $ or $ \mathbb{C} $.
Keith Conrad's webpage has an article on this although the Archimedian case is not proved there: http://www.math.uconn.edu/~kconrad/blurbs/gradnumthy/ostrowskinumbfield.pdf