We say that a field K is PAC if every variety absolutely irreducible over K has a K-rational point. Recall that a variety is absolutely irreducible if it is irreducible over overline{K}, and a K-rational point is an element on the variety with all its coordinates in K.

Amy Bright

Amy Bright

Answered question

2022-11-08

I am reading Fried-Jarden's book Field Arithmetic and I have a question on Pseudo Algebraically closed (PAC) fields.
We say that a field K is PAC if every variety absolutely irreducible over K has a K-rational point. Recall that a variety is absolutely irreducible if it is irreducible over K ¯ , and a K-rational point is an element on the variety with all its coordinates in K.
My question is the following: Is Q a PAC field? And more generally, is any number field PAC?
My attempt: I would like to find a polynomial in two variables with coefficients in Q which is irreducible over C and that does not have any root in Q 2 . This would show that Q is not PAC, which is my guess, but so far i've been unable to find this example.
I am also guessing that, more generally, every number field is not PAC, although this question is much more difficult...

Answer & Explanation

ontzeidena8a

ontzeidena8a

Beginner2022-11-09Added 17 answers

Explanation:
I think that the projective conic with equation
X 2 + Y 2 + Z 2 = 0
is absolutely irreducible over Q.

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