Lang 1952 paper terminology question theorem 12

Question

I am quoting here Theorem 12 from Lang's paper (page 387), which states the following:

Theorem. The following fields and their algebraic extensions are $C_1$:

1. The absolutely algebraic subfield of the maximal unramified extension of an ordinary $p$-adic field.

2. The convergent power series over an algebraically closed constant field.

The purpose of the post is not to find a proof of the theorem (which Lang does not provide), but actually to understand the terminology of its statement.

For example, it is clear what is the maximal unramified extension of an ordinary $p$-adic field, but is its absolutely algebraic subfield?

Also, what is the definition of the object in part 2.?

I struggled to find any answers for this on my own, so I hope somebody can clear me up.