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Linear Algebra proofs

  1. Apr 27, 2005 #1
    Linear Algebra proof

    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)
    Last edited: Apr 27, 2005
  2. jcsd
  3. Apr 27, 2005 #2
    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 [tex]\sum_{j=n}^\infty a_j p^{-j} [/tex] with [tex]0 \le a_j < p [/tex] for some [tex]n\in \mathbb{Z}[/tex]. 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.
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