# Tagged Questions

Homework questions are welcome as long as they are asked honestly, explain the problem, and show sufficient effort. Please do not use this as the only tag for a question. For the answers on homework questions, helpful hints or instructions are preferred to a complete solution. Please do not add ...

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### Computing the Chern-Simons invariant of SO(3)

I am an undergraduate learning about gauge theory and I have been tasked with working through the two examples given on pages 65 and 66 of "Characteristic forms and geometric invariants" by Chern and ...
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### Problem in Hamiltonian system

Not sure if this is too much physics to be here... Consider $$H:\mathbb{R}^{2N+1}\rightarrow\mathbb{R}$$ of class $C^2$, let $H(x,y,z)$ such that $x\in\mathbb{R}^N$, $y\in\mathbb{R}^N$ and ...
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### Number of automorphisms of saturated models

I have the following assignment question: Let $M$ be an $L$-model of cardinality $\kappa$. Assume $M$ is saturated. How can you show that $|\text{Aut}(M)|=2^{|M|}$? I see two possible ...
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### Rotman's exercise 2.8 “$S_n$ cannot be imbedded in $A_{n+1}$”

This question is about the (in)famous Rotman's exercise 2.8 in "An Introduction to the Theory of Groups" I've search and found similar questions here and in MO, but none of them contains a valid ...
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### Class group of $k[x,y,z,w]/(xy-zw)$

I had a homework problem (II.6.5 in Hartshorne) to compute the (Weil divisor) class group of $X=\operatorname{Spec} k[x,y,z,w]/(xy-zw)$. I have accomplished this; however, I used some results I don't ...
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### Polynomials with rational zeros

Find all polynomials $F(x)={a_n}{x^n}+\cdots+{a_1}x+a_0$ satisfying $a_n \neq0$ $(a_0, a_1, a_2 ... a_n)$ is a permutation of $(0, 1, 2 ... n)$ all zeros of $F(x)$ are rational
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### Soberification of a topological space

In Johnstone´s Stone Spaces, he introduces the concept of soberification of a topological space: Let $X$ be a topological space and $\Omega(X)$ the lattice of open subsets of $X$, the ...
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### On an exercise from Hartshone's Algebraic Geometry, Ch I sect 4

My question is about the Ex. 4.9 page 31 in the book GTM52 by Robin Hartshone. Let $X$ be a projective variety of dimension $r$ in $\mathbb{P}^n$, with $n\geq r+2$. Show that for suitable choice ...
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### An example of a compact multiplicatively unbounded ring

My teacher asked me to build an associative topological Hausdorff compact ring $R$ with $1$, which is multiplicatively unbounded. That means there is a neighborhood $U\ni 1$ such that $FU\not=R$ for ...
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### Probability, choose a box and then take exactly two white balls

There are $5$ boxes. There are $5$ white and $3$ black balls in two boxes, and $4$ white and $6$ black balls in the other three boxes. One box is randomly chosen. $3$ balls are randomly taken from ...
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### Way to Tietze's Transformation Theorem

during our knot-theory lecture we have talking about the following theorem: Given two finite presentations of the same group, one can be obtained from the other by a finite sequence of Tietze ...
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### cohomological proof of Maschke's theorem

I have been working on the following problem.. I have spent plenty of time trying to solve it myself. I am, however, unable to prove one small step in the argument. Beneath you can find my attempt. ...
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### Galois closure of a $p$-extension is also a $p$-extension

I'm working on a problem in Dummit & Foote and I'm quite stumped. The problem reads: Let $p$ be a prime and let $F$ be a field. Let $K$ be a Galois extension of $F$ whose Galois group is a ...
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### Complicated exercise on ODE

I have this exercise extracted from a examination of qualitative theory of ODE (in which we study the existence and uniqueness of solutions, and stability using the function of Liyapunov) I don't ...
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### Example of a ring satisfying this variant definition of “symmetric” on nilpotent elements

I want an example to show that if $a,b$ are nilpotent elements of a ring $R$ with 1 and if $c$ is any element of $R$, then $abc=0\Rightarrow acb=0$ but $cab=0$ does not imply $acb=0$. This is unlike ...
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### How does the fundamental theorem of algebra follow from Weierstrass’s theorem.

Could anyone please explain to me how the fundamental theorem of algebra follows from Weierstrass’s theorem (Bolzano Weierstrass)?
I am trying to assess the (Liapunov) stability of the equilibrium at $(0,0)$ of the system \begin{align*} x_1' &= -4x_2 + x_1^2 \\ x_2' &= 4x_1 + x_2^2. \end{align*} I plotted the phase ...
Question: Let $f:[a,b] \to \Bbb R$ be a continuously differentiable function s.t $f(a)=f(b)=0$ Prove that exists a point $c \in (a,b)$ such that $$|f'(c)| \ge \frac 4{(b-a)^2} \int ^b_a f(x) dx$$ ...