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

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- March 7th 2009, 12:07 PMdori1123prime subfieldLet be a field and let be the prime subfield of . Prove that every automorphism of fixes .
- March 7th 2009, 03:04 PMThePerfectHacker
- March 7th 2009, 06:35 PMdori1123
- March 7th 2009, 06:50 PMThePerfectHacker
- March 7th 2009, 07:23 PMdori1123
Yes.

- March 7th 2009, 07:58 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.