1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Automorphisms of Algebraic Numbers

  1. Nov 13, 2011 #1

    jgens

    User Avatar
    Gold Member

    1. The problem statement, all variables and given/known data

    1) Find more than two automorphisms of A.
    2) Do automorphisms of C fix AR?

    2. Relevant equations

    N/A

    3. The attempt at a solution

    I managed to figure out the second question since a map which preserves the additive structure of C will fix Q. And since the maps preserves multiplicative structure as well, it is not difficult to show that the automorphism will fix AR. So I think I have this part covered.

    So right now, I could use some help managing the first part of this problem. My professor indicated that there are a lot of automorphisms of A, so I'm thinking that there might be a family of automorphisms which aren't too difficult to construct. Any pointers on how to get started with this part are appreciated.

    Thanks!
     
    Last edited: Nov 13, 2011
  2. jcsd
  3. Nov 13, 2011 #2

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    Can you find an automorphism [itex]\mathbb{Q}[\sqrt{2}]\rightarrow \mathbb{Q}[\sqrt{3}][/itex]?? Can you extend this to [itex]\mathbb{A}[/itex]??
     
  4. Nov 14, 2011 #3

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Are you sure? Q[sqrt(2)] has an element such that x^2=2. Q[sqrt(3)] doesn't. I only know two automorphisms of A. Am I confused? I know lots of automorphisms of subfields of A. But not of A.
     
  5. Nov 14, 2011 #4

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    Ah yes :blushing:

    Maybe you can do something like [itex]\sqrt{2}\rightarrow -\sqrt{2}[/itex]??
     
  6. Nov 14, 2011 #5

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Q[sqrt(2)]=Q[-sqrt(2)]. Need to do better than that.
     
  7. Nov 14, 2011 #6

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    Yes, but it will be a nontrivial automorphism [itex]\mathbb{Q}[\sqrt{2}]\rightarrow \mathbb{Q}[\sqrt{2}][/itex]. Or am I just too tired??
     
  8. Nov 14, 2011 #7

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    True, I think you are right. But how to extend that to A? Maybe I'm too tired as well. Like I said before, I know lots of automorphisms of subfields of A. But the full field A? zzzzzzz!
     
  9. Nov 14, 2011 #8
    Haha. I am totally in your class. Math 257 with PSally?
     
  10. Nov 14, 2011 #9

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    Well you got the following theorem:

    If [itex]\phi:F_1\rightarrow F_2[/itex] is an isomorphism between fields and if [itex]F_i\subseteq K_i[/itex] is the algebraic closure, then [itex]\phi[/itex] can be extended to an isomorphism [itex]K_1\rightarrow K_2[/itex].​

    This theorem is true, but its proof is nonconstructive: it uses the lemma of Zorn. So you can show that there exists nontrivial automorphisms of A, but you can't construct one explicitly. :frown:

    I think this provides the OP with enough hints. So he should try to figure out the details now.
     
  11. Nov 14, 2011 #10

    jgens

    User Avatar
    Gold Member

    Yep.

    Indeed! Thanks!
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook