Results 1 to 5 of 5

Math Help - Algebra

  1. #1
    Junior Member
    Joined
    Mar 2009
    Posts
    27

    Algebra

    Let
    f(x) = x^4+ax^3+bx^2+cx+d
    be a polynomial in \mathbb Q[x] which has no rational root.

    Show that  f(x)  is irreducible in  \mathbb Q [x]

    if the cubic resolvent
     g(y)=y^3-by^2+(ac-4d)y-a^2d+4bd-c^2  has no rational root.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    10
    Quote Originally Posted by GaloisGroup View Post
    Let
    f(x) = x^4+ax^3+bx^2+cx+d
    be a polynomial in \mathbb Q[x] which has no rational root.

    Show that  f(x)  is irreducible in  \mathbb Q [x]

    if the cubic resolvent
     g(y)=y^3-by^2+(ac-4d)y-a^2d+4bd-c^2  has no rational root.
    If the cubic resolvent has no rational roots then it is irreducible. Therefore, the Galois group of f(x) must contain A_4 and so it splitting field is at least a 12-th degree extension. Now if the original polynomial was reducible then its will factor as a product of two quadradics which would mean its Galois group would be at most a 4 degree extension. Which is a contradiction.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Mar 2009
    Posts
    27
    Quote Originally Posted by ThePerfectHacker View Post
    Therefore, the Galois group of f(x) must contain A_4
    How can you conclude that the galois group contain A_4? what is the reason for it?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    10
    Quote Originally Posted by GaloisGroup View Post
    How can you conclude that the galois group contain A_4? what is the reason for it?
    I think that is a theorem about resolvents. If you never seen it then I do not know of anything else to say.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Junior Member
    Joined
    Mar 2009
    Posts
    27
    do you remember where you have seen this theorem in which book.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 1
    Last Post: February 4th 2011, 09:39 AM
  2. Replies: 2
    Last Post: December 6th 2010, 04:03 PM
  3. Algebra or Algebra 2 Equation Help Please?
    Posted in the Algebra Forum
    Replies: 4
    Last Post: May 12th 2010, 12:22 PM
  4. Replies: 0
    Last Post: April 24th 2010, 12:37 AM
  5. algebra 2 help
    Posted in the Algebra Forum
    Replies: 3
    Last Post: January 4th 2009, 07:24 PM

Search Tags


/mathhelpforum @mathhelpforum