All Questions
Tagged with model-theory fields
40
questions
20
votes
1
answer
802
views
Is there a minimal (least?) countably saturated real-closed field?
I heard from a reputable mathematician that ZFC proves that there is a minimal countably saturated real-closed field. I have several questions about this.
Is there a soft model-theoretic construction ...
- 230k
21
votes
1
answer
1k
views
Are the real numbers isomorphic to a nontrivial ultraproduct of fields?
Let $K_1, K_2, \dots$ be a countable sequence of fields, and let $\prod_{\mathcal F} K_i$ be the ultraproduct with respect to some nonprincipal ultrafilter $\mathcal F$.
Question: Can there be a field ...
- 62.6k
0
votes
0
answers
104
views
Can a definable group of definable automorphisms of a field contain the Frobenius automorphism?
Let $K$ be an infinite definable field of characteristic $p >0$ in a certain theory $T$ with a definable group of definable automorphisms. Can this group contain the Frobenius automorphism?
- 1
5
votes
1
answer
479
views
General algebraic result obtained from consideration on $\mathbb{Q}_p$
There are results in field theory which are obtained from, let's say, the complex numbers and then generalized to all algebraically closed fields.
For instance, the fact that a polynomial $P$ admits a ...
- 231
2
votes
1
answer
208
views
Minor Lefschetz principle
I once read (I think) the following equivalent formulation of the Minor Lefschetz principle:
If an elementary sentence holds for one algebraically closed field,
then it holds for every algebraically ...
- 4,503
2
votes
1
answer
237
views
Example of a bounded imperfect PAC field that is not separably closed
How do you construct a bounded (meaning there are only finitely many separable finite extensions of any given degree) imperfect pseudo-algebraically closed field that is not separably closed? I assume ...
- 311
10
votes
1
answer
301
views
How hard must "no high-degree irreducibles" proofs be?
Let $\mathsf{RCF}$ be the usual theory of real closed fields and for $n>2$ let $\theta_n$ be the sentence "No degree-$n$ polynomial is irreducible." Since $\mathsf{RCF}$ is complete, for ...
- 20.7k
12
votes
0
answers
515
views
Why is it so hard to give examples of differentially closed fields?
The theory of algebraically closed field, say in characteristic zero, and of differentially closed fields (of characteristic zero) have much in common: quantifier elimination and (hence) decidability; ...
- 30.8k
7
votes
0
answers
98
views
Reduced power of an ordered field
Suppose $K$ is an ordered field, $X$ is any set, and $F$ is a filter over $X$. Let $G$ be the reduced power of $K$ by $F$. That is, we take all functions from $X$ to $K$, then take equivalence ...
- 18.1k
4
votes
0
answers
197
views
Are there any 1-decidable algebraic extensions of $\mathbb{Q}$ which are not decidable?
A model $M$ is decidable if the set of all first-order formulas which are true in $M$ is a recursive set. And a model is $1$-decidable if the set of all existential formulas which are true in $M$ is ...
- 4,527
4
votes
1
answer
284
views
What is the lowest complexity definition of $\mathbb{Z}$ in an infinite algebraic extension of $\mathbb{Q}$?
In 2009, Jochen Koenigsmann showed that $\mathbb{Z}$ is universally definable in the field $\mathbb{Q}$. And in 2012, Jennifer Park proved a result which implies that $\mathbb{Z}$ is $\exists\forall$-...
- 4,527
27
votes
6
answers
2k
views
Is this theory the complete theory of the real ordered field?
We know that the real ordered field can be characterized up to isomorphism as a complete ordered field. However this is a second order characterization. That raises the following question. Consider ...
- 2,063
8
votes
1
answer
247
views
What is the lowest complexity definition of $\mathbb{Z}$ in an infinite extension of $\mathbb{Q}$
In 2009, Jochen Koenigsmann showed that $\mathbb{Z}$ is universally definable in the field $\mathbb{Q}$. And in 2012, Jennifer Park proved a result which implies that $\mathbb{Z}$ is $\exists\forall$-...
- 4,527
5
votes
1
answer
408
views
Is there a complete characterization of ordered fields without definable proper subfields?
$\mathbb{Q}$ has no proper subfields. As a result, all ordered fields elementarily equivalent to $\mathbb{Q}$ have no proper subfields which are first-order definable without parameters. And by the ...
- 4,527
4
votes
0
answers
186
views
Issue with "definition" of pseudo algebraically closed fields
I'm having an issue with a sentence in Chapter 11 of Fried & Jarden's Field Arithmetic. As a "motto" for pseudo algebraically closed (PAC) fields, they say they are fields $K$ such that &...
- 477