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

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

Hurkyl
Staff Emeritus
Gold Member
Well, does it satisfy the axioms of a field? Of a ring?

Hint: ( what is the zero element? ) (highlight to see)

mathwonk
Homework Helper
2020 Award
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.

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."

Best Regards,

Edwin

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

mathwonk
Homework Helper
2020 Award
what a great questiuon! i hVE NO IDEA!

mathwonk
Homework Helper
2020 Award
of course for endomorphisms we hVE ddition, multiplication and compositiopn, thTS THREE OPERtions.

Hurkyl
Staff Emeritus
Gold Member
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.

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.

Last edited: