1. ## I hAVE No clue wat the questions askin

Hi, This is one of my hw problem, N I just got no clue wat it is askin:

Suppose G: R to R is a field isomorphism, i.e. is a 1-1 and onto function such that G(x+y)=G(x)+G(y) and G(xy)=G(x)G(y) for all x and y belongs to R.
Prove that G preserves order. Deduce that G is the identity function.

....

Hi, This is one of my hw problem, N I just got no clue wat it is askin:

Suppose G: R to R is a field isomorphism, i.e. is a 1-1 and onto function such that G(x+y)=G(x)+G(y) and G(xy)=G(x)G(y) for all x and y belongs to R.
Prove that G preserves order. Deduce that G is the identity function.

....
By order, do you mean the order of the field, or some order on the elements of the field? The former is quite simple - it is a bijective function, and bijections preserve order! If it is the latter you mean, do we not need to be working in an ordered field? Fields, as far as I know, do not necessarily have any ordering of their elements.

The complex numbers have an automorphism ($\displaystyle G$ is an automorphism of $\displaystyle R$) that is not the identity function - complex conjugation.

I believe you will get more response to your posts in future if you type properly, without silly abbreviations. If you show people you have taken the time to ask your question by typing it out fully and checking it for errors, etc, then they will be more likely to take the time to answer you.

3. I would assume that the "R" in question is the field of real numbers with the usual order. In that case, it would be the latter of Swlabr's two possibilities and we are looking at an ordered field.

You might try looking at what such an isomorphism does to 0, 1, and -1 and their ordering.

4. Originally Posted by HallsofIvy
I would assume that the "R" in question is the field of real numbers with the usual order. In that case, it would be the latter of Swlabr's two possibilities and we are looking at an ordered field.

You might try looking at what such an isomorphism does to 0, 1, and -1 and their ordering.
Oh - I assumed R was for ring.

5. Originally Posted by Swlabr
Oh - I assumed R was for ring.
Good point, but the first post said "field isomorphism".