Norm of a Vector in a Number Field (or Order in a Number Field) I am looking for a measurement, which gives a length of a vector in a number Field? Is there any way or definition for that. 
For the Maximal order, What if, I tried to define a map from Maximal order to a Lattice. Is there any way to define a isomorphism from a order of a number field to a lattice. Please give me a example or a reference.
 A: A standard way to measure the size (= complexity) of vectors with coordinates in $\overline{\mathbb Q}$ is through the theory of Weil height functions. You can find the definition and properties of the Weil height
$$ H : \mathbb P^n(\overline{\mathbb Q}) \longrightarrow [1,\infty) $$
in many places, such as [1] and [2]. Since you are asking about vectors, rather than points in projective space, you probably want to use the embedding
$$ \overline{\mathbb Q}{}^n \longrightarrow \mathbb P^n(\overline{\mathbb Q}),\qquad
(a_1,\ldots,a_n) \longmapsto [1,a_1,\ldots,a_n] $$
and then take the height.
Among the many nice properties of $H$ is the fact that for any given number field $K$, the number of points in $\mathbb P^n(K)$ with height $H(P)\le C$ is finite. There's even an estimate for the number of such points as $C\to\infty$ due to Schanuel.


*

*Fundamentals of Diophantine Geometry, S. Lang, Springer, 1983.

*The Arithmetic of Elliptic Curves, J.H. Silverman, Springer, 2009.


Addendum to Answer OP's Comment: Let $A$ be a matrix with entries in a number field $K$, and let
$\boldsymbol{a}_1,\ldots,\boldsymbol{a}_n$ be its column vectors. We define the absolute value of a vector to be the max of the absolute values of its coordinates. Then the
following should be more-or-less correct, although one has to be a bit
careful about the normalizations of the absolute values.
$$
\begin{aligned}
  H(\det A) &= \prod_{v\in M_K} \max\bigl\{ \|\det A\|_v, 1 \bigr\}\qquad\text{(definition of $H$)} \\
  &\le \prod_{v\in M_K} \max\bigl\{ \|\boldsymbol{a}_1\|_v\cdot\|\boldsymbol{a}_2\|_v\cdots\|\boldsymbol{a}_n\|_v, 1 \bigr\}
    \qquad\text{(Hadamard)} \\
  &\le \prod_{v\in M_K} \prod_{i=1}^n \max\bigl\{ \|\boldsymbol{a}_i\|_v, 1 \bigr\}  \\
  &= \prod_{i=1}^n H( \boldsymbol{a}_i)\qquad\text{(definition of $H$)}.
\end{aligned}
$$
