# 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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### 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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### 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 searched and found similar questions here and in MO, but none of them contains a valid ...
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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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### 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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### 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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### Two parallel lines and distance between three points on them.

I'm solving one hard problem in my homework textbook (it is from the list of hardest problems in the end of book with stars). I reduced it to very simple question which I can prove by two, but very ...
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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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### 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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### Prove that (M,+,*) is a field

Prove that the multiplication $*:M \times M \to M$ defined by this table: * | 0 1 -------- 0 | 0 0 1 | 0 1 together with the commutative group (M,+), is a field (M,+,*). Group axioms: 1) ...
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### Non-Isomorphic Group Extensions

This is a question from a problem set on group cohomology, a subject I've just begun to learn. Let $B$ be a finite group and $A$ be abelian. I am looking for two groups $G_1$ and $G_2$ such that ...
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### Trying to solve $\lambda^3 - 3.250\lambda^2 + \lambda - 0.063 = 0$ using Newton-Raphson method

This is what I've atempted so far in solving $\lambda^3 - 3.250\lambda^2 + \lambda - 0.063 = 0$. The following are the steps: step 1: $f(\lambda) = \lambda^3 - 3.250\lambda^2 + \lambda - 0.063$ ...
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### Prove $(\mathbb Z \times \mathbb Z, \Sigma)$ to be a partial order and tell if its subset $T'$ is a lattice

Let $T = (\mathbb Z\times\mathbb Z, \Sigma)$ be defined as follows: \begin{aligned} (a,b) \text{ } \Sigma \text { } (c,d) \Leftrightarrow (a,b) = (c,d) \text{ or } a^2b^2<c^2d^2\end{aligned} ...
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### Digital Numbers using all digits from 1-9

Call a number a digital number if it consists of all the digits from 1-9, each used exactly once. What is the probability that a digital number will be divisible by 7 ? What is the probability that a ...
I think most people know these numbers. Find $x,\ y,\ z,\ w$ such that $x^3 + y^3 = z^3 + w^3$ and $x,\ y,\ z,\ w$ are not equal to each other. The first is $1729$. I'm trying to figure out if ...
Problem 24 of section 2 of Noncommutative Algebra by Farb & Dennis states: Let $R$ be an artinian ring and let $G$ be a finite group. Show that $R[G]$ is semisimple if and only if $R$ is ...