Linear Algebra proofs

[blaster]

I would appreciate any help with any of the foolowing:

1. Let C be a countable set. Prove that any linear well-ordered on C with the property that whatever c in C there are only finitely elements c` with c`<c, is unduced from the canonical order on N via a bijection N-> C. (N - natural no)

2. Prove that all algebraic numbers (all roots of polynomials with rational coefficients) form a countable subfield of C (complex).

3. Find a representation of the ring C[[x]] as an inverse limit of rings which have finite dimension over C.(complex).

4. Prove that the field Qp is uncountable. (Qp=field of fractions of Zp)

[snoble]

Number 4 is straight forward. You just need a surjection onto the reals (well in this case the non-negative reals which are also uncountable).

Hint. Given a number p any real number can be written in the form \sum_{j=n}^\infty a_j p^{-j} with 0 \le a_j < p for some n\in \mathbb{Z}. Do you know any representations of Qp that is similar? The surjection doesn't not need any homomorphic properties.

Easier than a diagonal argument. Though there is also a diagonal argument.