Finite and infinite primes of a number field Let $K$ be a number field. How to find the finite primes of a number field. can any one give some illustrations.  
 A: As I said, finding the absolute values for a number field corresponds to finding the primes for the corresponding number rings, and the embeddings of the number field into $\mathbb{C}$. 
You can find a proof of that here.
Finding all of them explicitly is a pain, and even when you can describe them all explicitly, it's not in a very enlightening form. Let me illustrate
For example, take $K=\mathbb{Q}(i)$. 
To find the Archimedean absolute values, we find the embeddings of $K$ into $\mathbb{C}$. They are, in particular $\sigma,\overline{\sigma}$ where $\sigma(a+bi)=a+bi$. In particular, define 
$$|a+bi|_\infty=|\sigma(a+bi)|=|\overline{\sigma}(a+bi)|$$
 where the absolute value on the RHS is the normal absolute value on $\mathbb{C}$. Then, $|\cdot|_\infty$ is the only infinite place of $K$. 
To find the finite places, we need to find the primes of $\mathbb{Z}[i]$. To do this, we use the Dedekind-Kummer theorem. Namely, let $p$ be a prime. Then, the primes above $p$ are $(p,g(i))$ where $g(x)$ is an irreducible factor of $x^2+1$ in $\mathbb{F}_p[x]$ (or, more appropriately, a lift of such a factor). But, we know that in $\mathbb{F}_p[x]$
$$x^2+1=\begin{cases}(x+1)^2 & \mbox{if}\quad p=2\\ \left(x+\alpha^{\frac{p-1}{4}}\right)\left(x-\alpha^{\frac{p-1}{4}}\right) & \mbox{if}\quad p\equiv 1\mod 4\\ x^2+1 & \mbox{if}\quad p\equiv 3\mod 4\end{cases}$$
Where $\alpha$ is a primitive root modulo $p$. Thus, the primes of $\mathbb{Z}[i]$ are 
$$P=\{(i+1)\}\cup\left\{\left(p,\pm\alpha^{\frac{p-1}{4}}\right):p\equiv 1\mod 4\text{ and }\alpha\text{ is a primitive root modulo }p\right\}\cup\{(p):p\equiv 3\mod 4\}$$
To each prime $\mathfrak{p}$ in this list, we can associate a valuation $v_\mathfrak{p}$ by defining $v_\mathfrak{p}(x)=m$ if $x\ne 0$ and 
$$(x)=\mathfrak{p}^m\cdot(\text{other primes besides }\mathfrak{p})$$
We can then extend this to a valution on all of $K$ in the natural way. This then induces an absolute value $|\cdot|_\mathfrak{p}$ in the usual way.
Thus, the Ostrowski theorem for number fields tells us that the places on $K$ are merely $|\cdot|_\infty$ and $|\cdot|_\mathfrak{p}$ for $\mathfrak{p}$ in $S$, and that this list is not redundant. 
Not any more satisfying than just the statement of Ostrowski, is it?
