# Questions tagged [abstract-algebra]

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57,554 questions
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### What are the patterns for these decompositions of permutations into transpositions?

By looking at answers on this site, in order to decompose a permutation into transpositions, so far I have seen 3 patterns as follows: (12345)=(15)(14)(13)(12) (12345)=(12)(23)(34)(45) (12)=(21) ...
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### What is the dicyclic group of order $12$? (What is $\mathbb{Z}_3\rtimes \mathbb{Z}_4$)

I have come across the dicyclic group of order $12$. I can see that this is generated by three elements subject to some relations. Is there a way to realize this group without talking about generators ...
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### Possibilities for a certain subgroup of $S_n$.

A two-part problem: $\bullet$ Let $H \leq S_n$ be the subgroup that fixes 1. Show that $H$ is isomorphic to $S_{n-1}$. $\bullet$ Show that there are no proper subgroups of $S_n$ that properly ...
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### If $a \in \mathbb{C}$ and $\exists n \in \mathbb{N}$ s.t. $\{ a^n, a^{n+1} \} \in \mathbb{N}$, prove $a \in \mathbb{N}$ [on hold]

Let $a$ be a complex number. If it exists a natural number $n$ (different of $0$), such that $a^n$ and $a^{n+1}$ are integers, prove that $a$ is an integer.
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### Localization and completion

Eisenbud, Exercise 7.1: Let $m$ be a maximal ideal of a ring $R$. Show that the map $R\to \hat{R}_m$ factors through the localization map $R\to R_m$. Here $\hat R_m$ is the completion of $R$ w.r.t....
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### Does transitivity of a relation imply associativity of an operation?

Let $*$ be a binary operation on a set $S$. We define a binary relation $R$ on $S$ by: $xRy$ iff $\exists z: x*z=y$. If $*$ is associative, then the relation $R$ is transitive. My question is whether ...
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### Why is m annihilating M/mM?

Let R be a Noetherian local ring with maximal ideal m. If M is an R-module, why is m the annihilator of M/mM?
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### some questions on finding the Galois group of the splitting field of $x^4-3$

I have two question regarding the Galois group of the splitting field of $x^4-3$. Firstly I know the roots of this polynomial are $\sqrt[4]{3},w\sqrt[4]{3},w^2\sqrt[4]{3},w^3\sqrt[4]{3}$, where $w$ ...
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### Prove that ((𝑁∩𝐻)𝑀)/𝑀 is a subgroup of 𝑁/𝑀. Then, Deduce ((𝑁∩𝐻)𝑀)/𝑀 is Abelian.

Suppose that 𝐻,𝑁,𝑀 are subgroups of 𝐺, 𝑀 is a normal subgroup of 𝑁. Assume 𝑁/𝑀 is Abelian. Prove that ((𝑁∩𝐻)𝑀)/𝑀 is a subgroup of 𝑁/𝑀. Then, Deduce ((𝑁∩𝐻)𝑀)/𝑀 is Abelian. I can ...
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### A question about the definition of path algebras

A quiver $Q=(Q_0,Q_1, s,t)$ is a quadruple consisting of a set $Q_0$ of vertices, a set $Q_1$ of arrows, and two maps $s,t :Q_1\rightarrow Q_0$ which associate each arrow $a\in Q_1$ to its source ...
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### Let 𝐿/𝑄 be a field extension. Show that complex conjugation defines an element of Aut𝑄(𝐿).

Let 𝐿/𝑄 be a field extension. Show that complex conjugation defines an element of Aut𝑄(𝐿). I am not sure about how to show it. Isn't it intuitive from the fundamental theorem of Galois theory?
Let $I = \{p(x) ∈ \Bbb Z[x]: 5\mid p(0)\}$. Prove that $I$ is an ideal of $\Bbb Z[x]$ by finding a ring morphism from $\Bbb Z[x]$ to $\Bbb Z_5$ with kernel $I$. Prove that $I$ is not a principal ideal....
### $R^{\mathbb N}$ as a free $R$-module.
Suppose that $R$ is a commutative ring. I'm wondering if the space $R^{\mathbb N}$ is a free $R$ module. I know how to prove that it is not a free $R$ module in the case of $R = \mathbb Z$. But the ...