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### Units of an extension of $\mathbb{Z}$ [closed]

Let $P(x)\in\mathbb{Z}[x]$ be monic and irreducible over $\mathbb{Q}[x]$, and let $\theta$ be a root of $P(x)$. Let $K = \{a + b\theta\} \subseteq \mathbb{Z}[\theta]$. When is it the case that there ...

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### “Degree 3 fields”

I was wondering what was known about fields $k$ having the property that any polynomial
over $k$ of degree $3$ has at least one root in $k$. Does such a field have a special name ?
Is there some kind ...

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### Group extensions isomorphic as groups

Let $G$ be a group and $A$ a $G$-module. It well know that there is a group isomorphism between the second cohomologoy group $H^2(G,A)$ and the abelian group $OpExt(G,A)$ of classes of extension ...

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### ${\rm Ext}^1$ and extensions of line bundles on a curve

I am confused about the following. I know that for two line bundles $L_1, L_2$ on an algebraic curve $C$ the vector space ${\rm Ext}^1(L_1,L_2)$ classifies isomorphism classes of rank two vector ...

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### Does every locally finite acyclic directed set embed into a linear order locally isomorphic to the integers? (Edit: extend, not merely embed.)

Let $S=(S,\prec)$ be a set together with an acyclic binary relation, generally nontransitive. $S$ is locally finite if, for every element $x\in S$, the sets $\{w|w\prec x\}$ ("direct past of $x$") ...

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### Does every countably infinite interval-finite partial order embed into the integers?

A partially ordered set $(S,\le)$ is called interval finite if the open intervals $(x,z):=\{y|x\le y\le z\}$ are finite for all choices of $x,z$ in $S$. An embedding $(S,\le)\rightarrow(S',\le')$ of ...

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### An isomorphism between different Ext's coming from group cohomology

Let $G$ be an abelian group and $M$ a $G$-module with trivial action. It is well-known that $H^2(G,M)$ classifies extensions of $G$ by $M$, which is $\mathrm{Ext}^1_{Ab}(G,M)$.
On the other hand ...

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### Can we extend a a.e. Lipschitz map defined on a closed subset of \bR^N to the whole space such that it is still a.e. Lipschitz

I have the following question: If $A$ is a metrically oriented $n$-dimensional subset of $\mathbb{R}^N$
and $f$ is a continuous map from $A$ to $\mathbb{R}^M$ . We know that $\mathrm{Lip} f < ...

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### Liftability of a submodule from an associated graded module

Let $k$ be a field, $A$ a $k$-algebra (probably noncommutative), and $M$ an $A$-module that's finite-dimensional as a vector space over $k$.
Let $Gr(M;k)$ denote the set of all $k$-subspaces of $M$, ...

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### Changing basis on an extension of a free Z-module.

Consider a finite-rank free $Z$-module $Y$. Let $c: Y \times Y \rightarrow Z$ be a $Z$-bilinear form. Assume that $c(y_1, y_2) + c(y_2, y_1)$ is even, for all $y_1, y_2 \in $. Then $c$ ...