Results 1 to 1 of 1

Math Help - Field Theory - Nicholson - Splitting Fields - Section 6.3 - Example 1

  1. #1
    Super Member Bernhard's Avatar
    Joined
    Jan 2010
    From
    Hobart, Tasmania, Australia
    Posts
    559
    Thanks
    2

    Field Theory - Nicholson - Splitting Fields - Section 6.3 - Example 1

    I am reading Nicholson: Introduction to Abstract Algebra, Section 6.3 Splitting Fields.

    Example 1 reads as follows: (see attachment)

    --------------------------------------------------------------------------------------------------

    Example 1. Find an extension  E \supseteq \mathbb{Z}_2 in which  f(x) = x^3 + x + 1 factors completely into linear factors.

    --------------------------------------------------------------------------------------------------

    The solution reads as follows:

    -------------------------------------------------------------------------------------------------

    Solution. The polynomial f(x) is irreducible over  \mathbb{Z}_2  (it has no root in  \mathbb{Z}_2  ) so

      E = \{ a_0 + a_1 t + a_2 t^2 \ | \ a_i \in \mathbb{Z}_2 , f(t) = 0 \}

    is a field containing a root t of f(x).

    Hence x + t = x - t is a factor of f(x)

    The division algorithm gives  f(x) = (x+t) g(x)   where  g(x) = x^2 + tx + (1 + t^2)

    , so it suffices to show that g(x) also factors completely in E.

    Trial and error give  g(t^2) = 0 so  g(x) = (x + t^2)(x + v)  for some  v \in F.

    ... ... etc (see attachment)

    -------------------------------------------------------------------------------------------------------------

    My problem is that I cannot show how  g(t^2) = 0 implies that  g(x) = (x + t^2)(x + v)  for some  v \in F.

    I would appreciate some help.

    Peter
    Attached Files Attached Files
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 1
    Last Post: October 18th 2013, 06:48 PM
  2. Replies: 6
    Last Post: October 15th 2013, 08:07 PM
  3. Replies: 6
    Last Post: October 15th 2013, 06:18 PM
  4. Replies: 3
    Last Post: October 14th 2013, 07:44 PM
  5. Field Theory - Nicholson - Section 6.2 - Exercise 31
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: October 8th 2013, 07:04 PM

Search Tags


/mathhelpforum @mathhelpforum