I have some algebra problems that I cannot solve and I'm hoping for some help.
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)
Well... there are some more but these are enough for now. Hope to get good complete answers as soon as possible. Thank you all.