Unanswered Questions
369,864 questions with no upvoted or accepted answers
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A proof of $\dim(R[T])=\dim(R)+1$ without prime ideals?
Background. If $R$ is a commutative ring, it is easy to prove $\dim(R[T]) \geq \dim(R)+1$, where $\dim$ denotes the Krull dimension. If $R$ is Noetherian, we have equality. Every proof of this fact I'...
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Is there a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?
Let $(X,\tau), (Y,\sigma)$ be two topological spaces. We say that a map $f: \mathcal{P}(X)\to \mathcal{P}(Y)$ between their power sets is connected if for every $S\subset X$ connected, $f(S)\subset Y$ ...
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Sorting of prime gaps
Let $g_i$ be the $i^{th}$ prime gap $p_{i+1}-p_i.$
If we rearrange the sequence $ (g_{n,i})_{i=1}^n$ so that for any finite $n$, if the gaps are arranged from smallest to largest, we have a new ...
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Does every ring of integers sit inside a ring of integers that has a power basis?
Given a finite extension of the rationals, $K$, we know that $K=\mathbb{Q}[\alpha]$ by the primitive element theorem, so every $x \in K$ has the form
$$x = a_0 + a_1 \alpha + \cdots + a_n \alpha^n,$$
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152
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Pullback and Pushforward Isomorphism of Sheaves
Suppose we have two schemes $X, Y$ and a map $f\colon X\to Y$. Then we know that $\operatorname{Hom}_X(f^*\mathcal{G}, \mathcal{F})\simeq \operatorname{Hom}_Y(\mathcal{G}, f_*\mathcal{F})$, where $\...
CommunityBot
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142
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Mondrian Art Problem Upper Bound for defect
Divide a square of side $n$ into any number of non-congruent rectangles. If all the sides are integers, what is the smallest possible difference in area between the largest and smallest rectangles?
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If polynomials are almost surjective over a field, is the field algebraically closed?
Let $K$ be a field. Say that polynomials are almost surjective over $K$ if for any nonconstant polynomial $f(x)\in K[x]$, the image of the map $f:K\to K$ contains all but finitely many points of $K$. ...
135
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Probability for an $n\times n$ matrix to have only real eigenvalues
Let $A$ be an $n\times n$ random matrix where every entry is i.i.d. and uniformly distributed on $[0,1]$. What is the probability that $A$ has only real eigenvalues?
The answer cannot be $0$ or $1$, ...
127
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On the Constant Rank Theorem and the Frobenius Theorem for differential equations.
Recently I was reading chapter $4$ (p. $60$) of The Implicit Function Theorem: History, Theorem, and Applications (By Steven George Krantz, Harold R. Parks) on proof's of the equivalence of the ...
CommunityBot
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Ring structure on the absolute Galois group of a finite field
Let $F$ be a finite field. There is an isomorphism of topological groups $(\mathrm{Gal}(\overline{F}/F),\circ) \cong (\widehat{\mathbb{Z}},+)$. It follows that the absolute Galois group carries the ...
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A question about divisibility of sum of two consecutive primes
I was curious about the sum of two consecutive primes and after proving that the sum for the odd primes always has at least 3 prime divisors, I came up with this question:
Find the least natural ...
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Classification of local Artin (commutative) rings which are finite over an algebraically closed field
A result in deformation theory states that if every morphism $Y=\operatorname{Spec}(\mathcal{A})\rightarrow X$ where $\mathcal A$ is a local Artin ring finite over $k$ can be extended to every $Y'\...
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What is the largest volume of a polyhedron whose skeleton has total length 1? Is it the regular triangular prism?
Say that the perimeter of a polyhedron is the sum of its edge lengths. What is the maximum volume of a polyhedron with a unit perimeter?
A reasonable first guess would be the regular tetrahedron of ...
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Why is a PDE a submanifold (and not just a subset)?
I struggle a bit with understanding the idea behind the definition of a PDE on a fibred manifold.
Let $\pi: E \to M$ be a smooth locally trivial fibre bundle.
In Gromovs words a partial differential ...
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Can Erdős-Turán $\frac{5}{8}$ theorem be generalised that way?
Suppose for an arbitrary group word $w$ ower the alphabet of $n$ symbols $\mathfrak{U_w}$ is a variety of all groups $G$, that satisfy an identity $\forall a_1, … , a_n \in G$ $w(a_1, … , a_n) = e$. ...
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