12
votes
3answers
434 views

Varieties where every algebra is free

I'd like to know more about varieties (in the sense of universal algebra) where every algebra is free. Another way to state the condition is that the comparison functor from the Kleisli category to ...
11
votes
1answer
425 views

First order decidability of rings vs Diophantine decidability

Are there known (preferably ``concrete'') examples of a ring $R$ (commutative, with 1) such that: $\bullet$ the first order theory of $R$ is undecidable, but $\bullet$ the positive existential (= ...
1
vote
0answers
157 views

What are the enforceable models of local artinian rings?

I was reading Hodges' "Model Theory" Chapter 8 a propos existentially closed models of $\forall_2$ theories in a countable first order language $L$. He extends the proof of the omitting type theorem ...
7
votes
0answers
227 views

Uncountable Lüroth problem

Question. Let $F(X)$ be the field obtained by adding an uncountable collection of indeterminates (mutually transcendental elements) to a prime field $F$. Is there an example of a subfield $E$ ...
2
votes
1answer
152 views

Terminology for system of equations and…

I am looking for the standard term for a system that consists of things of the form $p_i(x_1,\ldots ,x_n)=0$ and of the form $q_j(x_1,\ldots,x_n)\neq 0$ with the $p_i$ and $q_j$ polynomials. I have ...
11
votes
3answers
1k views

Intuition for Model Theoretic Proof of the Nullstellensatz

I recently read the model-theoretic proof of the Nullstellensatz using quantifier elimination (see www.msri.org/publications/books/Book39/files/marker.pdf). I'm convinced that the Nullstellensatz is ...
4
votes
1answer
300 views

Prime-ness checking for polynomial ideals over ACFs( algebraically closed fields).

Let $f_1,\ldots f_m \in k[X]$ have degrees bounded by $l$. and $I(\bar{f})$ be the ideal generated by $\bar{f}$. If $I(\bar{f})$ is not a prime ideal then its non-primeness is witnessed by polynomials ...