Assume DC($\aleph\_1$).


Can we define the following: 


1. Basis for a vector space $V$ over a field $K$ such that $\operatorname{card}(K) \leq \aleph\_1$ and we happen to find a generating set of $V$ of cardinality $\leq \aleph\_1$.
2. Linear dimension making the same assumption as above.
3. Transcendence degree of a field $K$ over $\mathbb{Q}$ if we happen to know that $K$ is of cardinality $\leq \aleph\_1$.
etc...



Some definitions without full choice
Without any appeal to the axiom of choice, if you have a vector space which is well-ordered then it has a basis. Moreover, if you have a generating set which is well-ordered then the vector space has a basis. The latter one allows you to have that the field itself is not necessarily well-ordered (as in your first point, where it is).


The same point applies to all three points. Once an object is well-ordered, we can use transfinite induction to do anything without any appeal to the axiom of choice.


(Do note, however, that in models like Solovay's model the cardinality of $\Bbb C$ is not $\aleph\_1$, and it may or may not have a transcendence basis over $\Bbb Q$.)


