Field automorphisms

It’s worth, I suppose, talking about automorphisms in general. An automorphism is a “renaming” of stuff so that it keeps the same structure. The “auto” means self. The “morph” means shape. The “same” is suppressed, because it really comes from the word “isomorphism,” which means same shape. So this is like a self-similar shape… but not in a fractally sense.

What is required for an automorphism? Well, any special constants need to stay the same. And the operations need to be “preserved,” whatever that means. For a field, it means that 0 and 1 are fixed, and that you can move + and \cdot outsied the function. More specifically, \phi is an automorphism of a field F if it is a function \phi:F\to F with

  • \phi(0)=0
  • \phi(1)=1
  • \phi(x+y)=\phi(x)+\phi(y)
  • \phi(xy)=\phi(x)\phi(y)
  • \phi is a bijection. That is, \phi doesn’t send two different points to the same point, and it sends something to every point. (Bijection is a fancy word meaning “1-1 and onto.”)
Can you think of any field automorphisms? In any field, you always have the trivial automorphism which sends a point x to x. It’s lame, but it counts.
What about something a bit more complicated? How about complex conjugation? What if \phi:\mathbb C\to \mathbb C by \phi(a+bi)=a-bi (for a,b\in \mathbb C)? Yes, this works. It’s not hard to check any of these rules, but I’m too lazy to do so.
How about something really crazy? Take the field \mathbb Q(\sqrt2), and the automorphism \psi defined by \psi(a+b\sqrt 2)=a-b\sqrt 2 (where a,b\in \mathbb Q). What? But that sends negative things to positive things? It doesn’t preserve the shape at all!
I assure you, it is a field automorphism. If you don’t believe me, check the rules yourself. The issue you’re potentially having, is that you’re thinking of \mathbb Q(\sqrt 2) as sitting inside \mathbb R, and thinking of the ordering on \mathbb R. But alas, I said nothing about order. It’s damn near impossible, but try to forget everything you know about the structure of \mathbb R, because it will only misguide you.
Advertisements

3 Responses to Field automorphisms

  1. Pingback: The group of field automorphisms « Andy Soffer

  2. Pingback: Automorphisms of Q « Andy Soffer

  3. Pingback: Fixed fields « Andy Soffer

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s