Let be a field and let be the prime subfield of . Prove that every automorphism of fixes .

Printable View

- March 7th 2009, 11:07 AMdori1123prime subfieldLet be a field and let be the prime subfield of . Prove that every automorphism of fixes .
- March 7th 2009, 02:04 PMThePerfectHacker
- March 7th 2009, 05:35 PMdori1123
- March 7th 2009, 05:50 PMThePerfectHacker
- March 7th 2009, 06:23 PMdori1123
Yes.

- March 7th 2009, 06:58 PMThePerfectHacker
The prime subfield is the intersection of all subfields i.e. it is the smallest subfield. If is a subfield then it means but then because is closed under addition and it has and it has all additive inverses. Therefore must be contained in . However, is a field and so for all . Therefore, the set of fraction is contained in , but this set of fractions is itself a field and so is in fact the set of all these fractions that we can form.