Automorphisms of Algebraic Numbers

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In summary, the conversation discusses finding automorphisms of A and whether they fix A∩R. The second question is solved by showing that an automorphism that preserves the additive and multiplicative structure of C will also fix A∩R. For the first question, the conversation suggests finding an automorphism \mathbb{Q}[\sqrt{2}]\rightarrow \mathbb{Q}[\sqrt{3}] and extending it to A, using the theorem that states an isomorphism between fields can be extended to their algebraic closures. However, this proof is nonconstructive, so the OP is advised to figure out the details on their own.
  • #1
jgens
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Homework Statement



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

Homework Equations



N/A

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!
 
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  • #2
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]??
 
  • #3
micromass said:
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]??

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.
 
  • #4
Dick said:
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?

Ah yes :blushing:

Maybe you can do something like [itex]\sqrt{2}\rightarrow -\sqrt{2}[/itex]??
 
  • #5
micromass said:
Ah yes :blushing:

Maybe you can do something like [itex]\sqrt{2}\rightarrow -\sqrt{2}[/itex]??

Q[sqrt(2)]=Q[-sqrt(2)]. Need to do better than that.
 
  • #6
Dick said:
Q[sqrt(2)]=Q[-sqrt(2)]. Need to do better than that.

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??
 
  • #7
micromass said:
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??

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!
 
  • #8
Haha. I am totally in your class. Math 257 with PSally?
 
  • #9
Dick said:
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!

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.
 
  • #10
MarqueeMoon said:
Haha. I am totally in your class. Math 257 with PSally?

Yep.

micromass said:
I think this provides the OP with enough hints. So he should try to figure out the details now.

Indeed! Thanks!
 

1. What is an automorphism of an algebraic number?

An automorphism of an algebraic number is a bijective function that preserves the algebraic structure of the number, meaning it maps algebraic operations (such as addition and multiplication) to their corresponding operations on the image number.

2. How do automorphisms relate to Galois theory?

Automorphisms play a significant role in Galois theory, as they are used to study the symmetries of algebraic equations. In fact, the fundamental theorem of Galois theory states that the automorphisms of a field extension correspond to the subgroups of the Galois group of that extension.

3. Can every algebraic number be expressed as an automorphism?

No, not every algebraic number has an automorphism. For example, irrational numbers such as π or √2 do not have any automorphisms, as they cannot be expressed as a root of a polynomial with rational coefficients.

4. What is the relationship between automorphisms and roots of polynomials?

One important relationship is that the roots of a polynomial correspond to the fixed points of an automorphism. In other words, if an automorphism maps a root of a polynomial to itself, then that root is a fixed point of the automorphism.

5. How are automorphisms used in cryptography?

Automorphisms are used in cryptography to help secure data by making it difficult to understand or decode. In particular, automorphisms are used in the construction of pseudorandom number generators, which are used to generate unpredictable numbers for cryptographic purposes.

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