Some definitions without full choice

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...
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(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 $\Bbb Q$.)
Thank you. But what is the cardinality of $\mathbb{C}$ in Solovay's model then? –  user38200 Aug 8 '13 at 11:51
In the Solovay model, the cardinality of $\Bbb C$ cannot be described by an ordinal. To see more about the continuum hypothesis itself, math.stackexchange.com/questions/404807/… –  Asaf Karagila Aug 8 '13 at 12:01
And also, is $\aleph_1$ the same thing as $\omega_1$ in these models? –  user38200 Aug 8 '13 at 12:06