is a number field by definition a subfield of $ \mathbb C $? I have seen that some authors are defing the number field as a subfield of $ \mathbb C$ which is a finite extension of the rational numbers $ \mathbb Q $, while some others without referering to complex numbers $ \mathbb C$ .
I think we don't need $ K$ to be a subfield of $ \mathbb C$ in the definition.
So, my question is the follwing:

Is it neceserily to define $K$ as a subfield of $ \mathbb C$ or not ?
  And if no why ? Is it true that if we omit this in the definition, that then $K$ will turn out to be a subfield of $ \mathbb C$?

I came up with this question, when I saw that in order to define infinite primes in a number field then these are determined by the embeddings $ K \to \mathbb C $
Any idea would be really appreciated.
Thank you in advance.
 A: Usually there will be more than one way to consider $K$ as a subfield of $\mathbb{C}$ - or more precisely, to embed $K$ in $\mathbb{C}$.
For example, take $K:=\mathbb{Q}[X]/(X^3-2)$, which is to say the extension field of $\mathbb{Q}$ of degree $3$ containig a root of the polynomial $X^3-2$ (which root we choose is irrelevant for now).
Then we can embed $K$ into $\mathbb{C}$ by mapping that fixed abstract root of the polynomial to the complex number $\sqrt[3]{2}$, which is also a real number, or the complex number $\sqrt[3]{2}\cdot \exp(\frac{2\pi i}{3})$, which is a non-real complex number.
So we have embedded $K$ into $\mathbb{C}$ in two disctinct ways. This distinction would have been "lost", had we considered $K$ as a subfield of the complex numbers from the start.
A: A number field has a purely abstract definition, however every one of them is isomorphic to a subfield of $\mathbb C$.  Does that answer your question?
A: A number field is a finite extension of $\Bbb{Q}$. As such, it is always isomorphic to a subfield of $\Bbb{C}$: there are exactly $[K:\Bbb{Q}]$ copies of $K$ in $\Bbb{C}$, corresponding to the different embeddings $K \to \Bbb{C}$.
While we can think of $K$ as an abstract field, e.g. as a quotient of $\Bbb{Q}[X]$ (such a thing cannot be contained in $\Bbb{C}$: why?), it can be useful to identify it with one of its copies in $\Bbb{C}$, for example to define an absolute value on $K$. Indeed, each pair of conjugate embeddings (or a single embedding, if its image is in $\Bbb{R}$) gives rise to a different absolute value on $K$, which extends the usual absolute value on $\Bbb{Q}$...

The above question was meant as a thought exercise, though maybe its formulation was a bit hasty. A quotient of $\Bbb{Q}[X]$ is a collection of subsets of $\Bbb{Q}[X]$, which are not elements of $\Bbb{C}$, at least with the usual definitions of $\Bbb{C}$.
True, we could define $\Bbb{C}$ as the completion of $K = \Bbb{Q}[X]/(X^2 + 1)$ with respect to an extension of $|\cdot|$ (in which case of course $K$ would be contained in $\Bbb{C}$), though I doubt that the OP is using this definition...
