Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Algebraic And Simple Extensions

  1. Sep 22, 2010 #1
    1. The problem statement, all variables and given/known data
    I'll be delighted to get an answer to the following question:
    Does every algebraic extension of a field is a simple extension?
    2. Relevant equations
    3. The attempt at a solution
    I'm pretty sure that the answer is negative... I was thinking on taking the field of all the algebraic numbers over [tex] Q [/tex] ... this field is obviously an algebraic extension of Q, but how can I prove it isn't simple? (I'm pretty sure that its degree is infinity, but have no idea how to to prove it) ...

    Hope you'll be able to help me

    Thanks!
     
  2. jcsd
  3. Sep 22, 2010 #2

    Office_Shredder

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    To prove the degree you can use the tower law. [tex]2^{1/n}[/tex] is obviously in the algebraic numbers for each n, so if A is the set of algebraic numbers

    [tex][A:Q]=[A:Q(2^{1/n})][Q(2^{1/n}):Q][/tex]
     
  4. Sep 24, 2010 #3
    Thanks a lot!
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook