Q is the prime subfield of any field of characteristic 0, proof that
Proposition. is the prime subfield of any field of characteristic 0.
Let be a field of characteristic . We want to find a one-to-one field homomorphism . For with coprime, define the mapping that takes into . It is easy to check that is a well-defined function. Furthermore, it is elementary to show
This shows that is a field homomorphism. Finally, if and , then , a contradiction. ∎
Proposition. () is the prime subfield of any field of characteristic .
Let be a field of characteristic . The idea again is to find an injective field homomorphism, this time, from into . Take to be the function that maps to . It is well-defined, for if in , then , meaning , or that , (showing that one element in does not get “mapped” to more than one element in ). Since the above argument is reversible, we see that is one-to-one.
To complete the proof, we next show that is a field homomorphism. That and are clear from the definition of . Additivity and multiplicativity of are readily verified, as follows:
This shows that is a field homomorphism. ∎
|Title||Q is the prime subfield of any field of characteristic 0, proof that|
|Date of creation||2013-03-22 15:39:57|
|Last modified on||2013-03-22 15:39:57|
|Last modified by||CWoo (3771)|