A quick Question about the set of Automorphisms of a field F

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In summary: Best Regards,In summary, the set of automorphisms of a field F does not form a field under point-wise addition and multiplication of functions. It is probably not closed under either addition or multiplication, but has not been checked. The underlying motivation for the development of ring theory may have been an attempt to answer questions about the integers that could not be answered by exploring systems "more inclusive" than the integers themselves.
  • #1
Edwin
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Does anyone know if Aut(F), the set of automorphisms of a field F, form a field under point-wise addition and multiplication of functions? If not, does it form a ring?

Inquisitively,

Edwin
 
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  • #2
Well, does it satisfy the axioms of a field? Of a ring?

Hint: ( what is the zero element? ) (highlight to see)
 
  • #3
the set of field automorhisms, say of a finite normal extension of Q of vector dimension n, forms a group of order n.

hence it cannot be field unless n is a power of some prime number.

it also cannot have a unit for multiplication, since that could only be the constant function 1, which being constant, is not an automorphism.

it is probably not closed under either addition or multiplication but i have not cgecked it.

try some simple examples, like a quadratic extension. if the group consistssay of id and conjugation, try adding z and zbar.
 
  • #4
That makes sense. I also posted the question on "yahoo answers" and someone pointed out that "if f is an automorphism, then so is -f and that f+(-f) = 0. 0 is not an not an automorphism so Aut(f) is not closed under point-wise addition of functions. So (Aut(F), +) can not be a group, and so can not be an abelian group. Hence (Aut(F), +, *) is not a ring and so can not be a field."


Thanks for your help guys!

Best Regards,

Edwin
 
  • #5
Does anyone know what the underlying motivation was to develop ring theory? I've heard various arguments stating that ring theory was developed in an attempt to answer questions about the nature of the integers by exploring systems "more inclusive" than the integers themselves. The reason I am asking this is I want to know whether there might be a motiviation to development an abstract theory with three binary operations that satisfy certain distributive laws. Why did mathematicians not develop more abstract theory for two, three, or n associative binary operations on a set? Is it just because no one has gotten around to it, or is it because there is good reason to believe that such theories would not be of much use? Like for example, the complexity of the theory makes the results of work too complicated to be of much practical or theoretical use.

Inquisitively,

Edwin
 
  • #6
what a great questiuon! i hVE NO IDEA!
 
  • #7
of course for endomorphisms we hVE ddition, multiplication and compositiopn, thTS THREE OPERtions.
 
  • #8
Edwin said:
Why did mathematicians not develop more abstract theory for two, three, or n associative binary operations on a set?
Why stop at binary operations, or associative ones? You might want to look into universal algebra. Or maybe model theory.
 
  • #9
I have at home a book called 'A Modern View of Geometry' (can't recall the author's name just now) in which the properties of projective spaces are gradually developed through a series of finite point models of the axioms. Hand in hand with this, the author sets up an algebraic structure, which functions as a sort of primitive Cartesian geometry. As more axioms are added, the algebra becomes richer and the proto-projective space takes on more of the characteristics we expect from a recognizable space.

The reason I mention this is that the fundamental algebraic object he uses is a TRINARY ring. This is the only place I have seen a trinary ring actually in use as an analytical tool, instead of being the object of the analysis itself.

As a side note, I noticed a very strange coincidence: the 7 point projective space mentioned in the above reference has the same structure as the multiplication table for the octonians, in the sense that each projective line contains exactly 3 points and there is an isomorphic mapping of the unit basis of the octonians to points in the projective space, such that the product of any two elements of the octonian basis (except 1) is precisely the third point on the line containing the two elements that were multiplied.

I don't know what to make of this, but I thought it was an interesting connection between octonians and projective geometry.
 
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1. What is the definition of an automorphism in the context of a field?

An automorphism of a field F is a bijective function from F to itself that preserves the field operations of addition and multiplication. In other words, an automorphism is a function that maps elements of the field to other elements in a way that the field properties remain unchanged.

2. How do automorphisms relate to the concept of symmetry in mathematics?

Automorphisms can be seen as a type of symmetry in mathematics because they preserve the structure of a field. Just like how a symmetrical object stays unchanged after a transformation, a field remains unchanged after being mapped by an automorphism.

3. Can all fields have automorphisms?

No, not all fields have automorphisms. For example, the field of real numbers has no non-trivial automorphisms, meaning that the identity function is the only automorphism of this field. However, some fields, such as finite fields, have a large number of automorphisms.

4. How are automorphisms of a field F related to the Galois group of F?

The automorphisms of a field F form a group called the Galois group of F. This group captures the symmetries of the field and is important in studying the solutions to polynomial equations over F.

5. Are automorphisms of a field F unique?

Yes, automorphisms of a field F are unique. This means that for a given field, there is only one way to map its elements while preserving the field operations. However, different fields may have the same automorphisms if they are isomorphic, meaning they have the same structure.

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