# Tagged Questions

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### Proves or counterexamples in retraction and coretractions of modules

Any tip for proving or counterexamples that the following morphism of $\mathbb Z$-modules ${\mathbb Z} \to {\mathbb Q}$ is not a retraction and ${\mathbb Q} \to {\mathbb Q/\mathbb Z}$ is not a ...
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### Irreducibility of $\frac{X^{p^n}-1}{X^{p^{n-1}}-1}$.

Exercise: Let $p$ be a prime number. Then, the polynomial $$\frac{X^{p^n}-1}{X^{p^{n-1}}-1}$$ is irreducible over $\mathbb Z[X]$, for any integer $n \geq 1$. I'm able to ...
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### Finite p-primary group question

Problem is rotman's Intrdouction to theory of group, Exercise 6.8. Let G be a direct sum of b cyclic groups of order $p^{m}$. If $n< m$, then $p^{n}G/p^{n+1}G$is elementary and ...
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### Elementary proof of irreducibility criterion

From Problems from the Book'' by Andreescu and Dospinescu, the following irreducibility criterion is presented: Let $f$ be a monic polynomial with integer coefficients and let $p$ be a prime. If ...
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### Equalities in Groups, How prove this? [duplicate]

Let $H,K≤ G$, for all $g ∈ G$ , $$\frac {|H||K|}{|H∩(gKg^{-1})|}= \frac {|H||K|}{|(g^{-1}Hg)∩ K|}$$ I try to show this but I do Know how to attack this exercise.
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### Solution Manual for Chapters 13 and 14, Dummit & Foote

I bought the third edition of "Abstract Algebra" by Dummit and Foote. In my opinion this is the best "algebra book" that has been written. I found several solution manual but none has solutions for ...
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### Finding an algebra of smooth functions on a manifold with a given product.

I am having trouble with the following exercise from Jet Nestruev's Smooth Manifolds and Observables. It is one of the first exercises in the book and I have no idea how to approach this type of ...
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### Nice exercises on resultants

I would like to ask if some one knows a source (a book, or lecture notes ect) that contains several nice exercises on resultants of polynomials (it would be nice if there were some solutions as well ...