# All Questions

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### How can I determine least common multiple for a given number and all numbers before it?

The wikipedia article on least common multiples only talks about determining the least common multiple between 2 numbers. I'm looking for an algorithm that will determine it for a set of numbers 1 .. ...
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### A Problem about Galois Extension

This is a basic question about Galois Extension, but I want some details about it. Let $F$ be a splitting field over $\mathbb Q$ the polynomial $x^8-5\in\mathbb Q[x]$. Recall that $F$ is the subfield ...
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### Find the antiderivative of the following problem

Find the antiderivative of $\pi(\frac{4}{y^2})$. This has to do with a volume problem. And I'm using the disk method to solve. So the pi needs to incorporated.
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### If I remove the premise $a\neq b$ in this question, will the statement still be true?

I encountered this proving problem, I can do the proof but my question is why in the condition/premise we need $a$ to be unequal to $b$? My guess is that even $a=b$, the statement is still true, is it ...
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### Behavior of the resolvent near the boundary of the spectrum

My question is, in some sense, a continuation of the question below. Isolated singularities of the resolvent Suppose $T\in B(H)$ has no eigenvalues, pick $x\in H$, $x\neq 0$, and consider the ...
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### Sylow $p$ subgroups of $S_{2p}$ and $S_{p^2}$ - Dummit foote - $4.5.45; 4.5.46$

Question is to find Sylow $p$ subgroups of $S_{2p}$ for odd prime $p$ and show that this is an abelian group of order $p^2$ Sylow $p$ subgroups of $S_{p^2}$ for odd prime $p$ and show that this is ...
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### Finding number of primes of the form $1010\ldots 101$ (in base 10) [closed]

How many primes among the positive integers, written as usual in the base $10$, are such that their digits are alternating $1$’s and $0$’s, beginning and ending with $1$?
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### Name for numbers expressible as radicals

What is the name (there must be one!) for real (or perhaps complex) numbers expressible as radicals? Radical numbers? Solvable numbers? (following the same logic as ‘solvable group’). In other ...
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### Number of intermediate fields in non-separable extensions that are also not purely inseparable

If a field extension is separable then we know there are only finitely many intermediate fields. If a field extension is purely inseparable then it is possible for there to be infinitely many ...
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### Is a simple curve which is nulhomotopic the boundary of a surface?

Let $C$ be a simple curve in an open subset $U$ of $\mathbb R^3$. Suppose that $C$ is nulhomotopic in $U$. Must there exist a homeomorphism $f$ from the closed unit disk $D$ in $\mathbb R^2$ to $U$ ...
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### Writing $x^2+y^2+z^2$ as a polynomial combination of $xyz$, $x+y+z$, and $\dfrac1x+\dfrac1y+\dfrac1z$

Can we write $x^2+y^2+z^2$ as a polynomial combination of $xyz$, $x+y+z$, and $\dfrac1x+\dfrac1y+\dfrac1z$? What about $x^3+y^3+z^3$?
If $f:\mathbb R\to \mathbb R$ is a continuous function satisfying $\vert f(x)-f(y)\vert\geq \dfrac{1}{2}\vert x-y\vert$ for all $x,y\in \mathbb R$, then is $f$ bijective? I believe that $f$ is ...
Let $f_1,...,f_N$ a collection of fucntions, $\epsilon_1,...,\epsilon_N$ randomized signs ( $\pm1$) with same probability and $N\in\mathbb{N}$. If \$ ...