1. ## Isomorphism of fields

Let φ: F1 --> F2 be an isomorphism field. Prove that φ(1) = 1. (That is, prove that φ must map the multiplicative identity of F1 to the multiplicative identity of F2.)

I know that there is a "1" element in F1 and a "1" element in F2, because they're fields. I also know that because φ is isomorphic, there exists a bijection from F1 to F2.
I need to prove that φ takes the "1" element in F1 to the "1" element in F2.

Can anyone help me from here? Thanks in advance.

2. Originally Posted by page929
Let φ: F1 --> F2 be an isomorphism field. Prove that φ(1) = 1. (That is, prove that φ must map the multiplicative identity of F1 to the multiplicative identity of F2.)

I know that there is a "1" element in F1 and a "1" element in F2, because they're fields. I also know that because φ is isomorphic, there exists a bijection from F1 to F2.
I need to prove that φ takes the "1" element in F1 to the "1" element in F2.

Can anyone help me from here? Thanks in advance.

As we've an isomorphism, $\displaystyle \forall b\in F_2\,\exists a\in F_1\,\,s.t.\,\,\phi(a)=b\,,\,\,1_2=\phi(a')$ , but then

$\displaystyle b=1_2\cdot b = \phi(a')\phi(a)=\phi(a'a)$ . Now use that the above a is unique (because of 1-1) and...

Tonio