Abstract algebra is the study of algebraic objects. Some of the more common algebraic objects are groups, rings, fields, vector spaces, modules, and other advanced topics.

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Is my understanding for group algebra correct?

Let $G$ be a group, $k$ be a commutative unital ring. Consider $\mathbf{Alg}_k^{inv}$ the category of unital $k$-algebras whose multiplicative semigroup is a group. Then there is a forgetful functor ...
9
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1answer
112 views

Product of numbers $\pm\sqrt{1}\pm\sqrt{2}\pm\cdots\pm\sqrt{n}$ is integer

Prove that the product of the $2^n$ numbers $\pm\sqrt{1}\pm\sqrt{2}\pm\cdots\pm\sqrt{n}$ is an integer. I want to consider the polynomial $P(x)=(x-a_1)(x-a_2)\cdots(x-a_{2^n})$, where the $a_i$'s are ...
6
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2answers
42 views

Gallian: is it true that the well-ordering principle can't be proven from properties of arithmetic?

I just started reading Gallian's Abstract algebra and on page 3 it says "An important property of the integers...is the so-called Well Ordering Principle. Since this property cannot be proved from the ...
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1answer
25 views

Showing that an epimorphism of an ideal is again an ideal

Let $R, S$ be commutative rings, $f : R \rightarrow S$ an epimorphism, I an ideal of R. Show that $f(I)$ is an ideal of $S$. As far as I understand, I need to show 4 things: 1) $0_s \in f(I)$ ...
0
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1answer
67 views

Is ideal prime or maximal? [on hold]

Find, whether or not given ideal of $\mathbb{Z}[x]$ ring is prime or maximal and describe the quotient ring : a) $J_1 = (x-5)$ b) $J_2 = (3, x+5)$. How can I do that?
2
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1answer
28 views

Example of finite ring with a non principal ideal

I would like to have an example (or a class of examples) of a finite ring (with unit, not necessarily commutative) that is not a principal ideal ring (if possible an example of a local ring and of ...
7
votes
1answer
111 views

What's the algebraic closure of the quaternions?

This post is a sequel of: Is the set of quaternions $\mathbb{H}$ algebraically closed? This answer shows that: 1. $\mathbb{H}$ is algebraically closed for the polynomials of the form $\sum a_r ...
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0answers
34 views

Determine $\operatorname{Hom}(\mathbb{Z},\mathbb{Z}/n)$ and $\operatorname{Hom}(\mathbb{Z}/n, \mathbb{Z})$. [on hold]

Determine $\operatorname{Hom}(\mathbb{Z},\mathbb{Z}/n)$ and $\operatorname{Hom}(\mathbb{Z}/n, \mathbb{Z})$, where $n$ is positive integer (as $\mathbb{Z}$-module). I can only find a ...
0
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1answer
31 views

Given the permutation, which symmetric group does it belong to?

Given the permutation (1, 2, 4)(3, 5, 6) is it clear which symmetric group this permutation belongs to? Explain. so from here I got: 1 2 3 4 5 6 2 4 5 1 6 3 and that is an element of S6 Im ...
0
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1answer
27 views

Express σ using matrix notation

Suppose σ = (3, 4, 5)(2, 4, 5) ∈ S5. (a) Express σ as the product of disjoint cycles. (b) Find the order of σ. (c) Is σ even or odd? (d) Express σ using matrix notation. (e) Find σ-1 Im not ...
0
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1answer
22 views

Free module over a ring with identity with a basis of size m, ∀m≥n

Please, help on this Exercise [Hungerford's Algebra, IV.2.12] If $F$ is a free module over a ring with identity such that $F$ has a basis of finite cardinality $n\geq 1$ and another basis of ...
0
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0answers
30 views

Example of a P.I.D. that is not a Euclidean domain? [duplicate]

What would be an example of a principal ideal domain that is not a Euclidean domain? What is a general strategy for proving this?
2
votes
2answers
66 views

What can we say about the order of a group?

Let $G$ be a group and $a ∈ G$. If $a^{12}= e$, what can we say about the order of $a$? What can we say about the order of $G$? We know that $|a|$ divides $12$, but what can we say about the order of ...
0
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2answers
35 views

If $G$ is abelian, prove that $H = \{g \in G \mid g^2 = e\}$ is a subgroup of $G$. [duplicate]

Let $G$ be an Abelian group. Prove that $H = \{g \in G \mid g^2 = e\}$ is a subgroup of $G$. I know something similar to this has been asked, but I just want to check my understanding/reasoning: We ...
0
votes
1answer
36 views

Is the factor group $D_n$/$C_n$ abelian? [on hold]

Looking for a solvable chain for $D_n$. $C_n$ is the normal subgroup of rotations in $D_n$. Thanks.
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2answers
43 views

$K \unlhd H\unlhd G$, K is sylow in H, proof $K \unlhd G$

$K \unlhd H\unlhd G$, K is sylow in H, proof $K \unlhd G$ I know in general it's false, I wonder how should I use the condition that K is sylow? is it that the K is the unique sylow subgroup? and ...
0
votes
1answer
25 views

Proving existence for a combination of mappings from a group to a commutative ring

Let $G$ be a group and $R$ be a commutative ring. Let $X$ be the set of all mappings $\phi : G \rightarrow R$ with $\phi(g) \neq 0$ for finitely many $g \in G$. For all $g \in G$ define $$(\phi_1 ...
0
votes
1answer
14 views

domain and range and determine whether * defines a binary operation on G?

Define the function $∗ : G × G → G$ by $∗(g_1, g_2) = g_1g_2$ where $G = \{0, 1\}$. Explicitly write out the domain and range of $∗$ and then determine whether $∗$ defines a binary operation on $G$. ...
0
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1answer
33 views

Proving relations between (sub-)rings and a group

Let $R \neq 0$ be a commutative ring, $G$ be a finite group, $\#G > 2$. 1) $H$ subgroup of G $\Rightarrow$ monoid ring R[H] is a subring of monoid ring R[G] 2) Let $x := \sum_{g\in ...
7
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2answers
76 views

Units of $\overline{\mathbb{Z}}$

What are the units of $ \overline{\mathbb{Z}} $ (the ring of algebraic integers)? I know all roots of monic polynomials with constant term 1 are units, but are there any others?
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3answers
51 views

Mapping Normalizer to Automorphism

I am squeezing my brain trying to understand this problem: Let H be a subgroup of $G$, and denoted by $\operatorname{Aut}(H)$ the group of all automorphisms of $H. $ Define $$\alpha : N_G(H) ...
0
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1answer
23 views

Show there exists a subgroup of order 15

I have a group $|G|=375=5^3*3$ by Sylow analysis, I have shown that $H_5$ is normal, but $H_3$ is not necessarily normal. My question is if I assume $H_3$ is normal, how do I show there is a subgroup ...
6
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1answer
68 views

Number theory and abstract algebra question

So I was solving this question Find an isomorphism from the additive group $\mathbb Z_6$ to the multiplicative group of units $U_7$ in $\mathbb Z_7$. I found that $3$ is generator for U7 by brute ...
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1answer
49 views

A question on Artinian and Noetherian rings.

All rings are commutative and unital. Suppose that $A$ is a ring in which the zero ideal can be written as a product of maximal ideals of $A$. I try to prove that $A$ is Noetherian if and only if ...
2
votes
1answer
28 views

Show this function is onto

I gave a mapping A to C such that A is the set of left cosets in G described as $A$={$N(H), gN(H),...g_nN(H)$} for N(H) is the normalizer of H in G and C is the set of conjugates of H, ...
5
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0answers
79 views

A More Advanced Version Of Aluffi

Paulo Aluffi's Book, Algebra, Chapter 0 aims to teach basic algebra from a categorical viewpoint. The first chapters of the book, however, introduce groups and rings using very basic categorical ...
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2answers
44 views

Isomorphism between two sets

Let $G$ be the additive group of all real numbers, and let $G_0$ be the group of all positive real numbers under multiplication. So I defined the following map $\phi(x) = 10^x$ and I proved its well ...
0
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0answers
31 views

If $H<S_n$, $H$ is abelian and transitive on $\{1,2,…,n\}$, then the order of $H$ is $n$ [duplicate]

If $H<S_n$, if $H$ is Abelian and transitive on $\{1,2,...,n\}$, then the order of $H$ is $n$. So far I have: $H$ is transitive therefore the group orbit is the group itself. I know I should ...
4
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1answer
71 views

A question on the relation of the Galois group as field automorphisms and the Galois group as permutations of roots

Let $f\in K[X]$ be a monic separable polynomial and $L$ a splitting field of $f$. Let $M=\{l_1,\ldots,l_n\}$ be the set of roots of $f$ in $L$, i.e. $$ f=(X-l_1)\cdots(X-l_n). $$ The Galois group ...
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2answers
51 views

Trying to understand a proof for the automorphisms of a polynomial ring

Consider the following proof for finding all automorphisms of the ring $\mathbb{Z}[x]$ which I am trying to understand. Source I have two question regarding the proof 1) They set $d = ...
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1answer
33 views

Finding the ideal

Determine all the ideals, prime ideals, and maximal ideals of $\mathbb{R}[x]/I$ where $I$ is the ideal generated by $(x^2+1)(x-2)^2$. I am currently doing some reading on ideals (see ...
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1answer
15 views

Finiteness of a simple extension

Here I have two propositions from p.521 on Abstract Algebra written by Dummit Foote. Let $\alpha$ be algebraic over the field $F$ and let $F(\alpha)$ be the field generated by $\alpha$ over $F$. ...
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0answers
66 views

The structure of the Sylow p-subgroups of $Sym(n)$

For the structure of the Sylow $p$-subgroups of $Sym(n)$, there is a standard proof by using the properties of the wreath product like as in the Passman's book. But I want to understand this proof. In ...
3
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0answers
22 views

Closed subset of a affine linear group [duplicate]

Let $G\subseteq GL_n(\mathbb{C})$ a Zariski-closed linear subgroup and $X\subseteq G$ closed with $X*X\subseteq X$ and $e \in X$. Then $X$ is a subroup. I am not sure how to start here. I know that ...
5
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1answer
37 views

Determine whether an ideal is principal or not

Let $I=\{a+b\sqrt{-3}: a+b \text{ even}\}$ be an ideal in $R=\mathbb{Z}[\sqrt{-3}]$. I want to determine whether $I$ is a principal ideal or not. I've been trying to work with the ideal $(2)$. I ...
1
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3answers
63 views

Is $\mathbb{Q}[x]/(x-1)(x^2+4)\mathbb{Q}[x]$ isomorphic to $\mathbb{Q}\times\mathbb{Q}[i]$?

Problem: Check if $\mathbb{Q}[x]/(x-1)(x^2+4)\mathbb{Q}[x]$ is isomorphic to $\mathbb{Q}\times\mathbb{Q}[i]$. If not, find what is it isomorphic to. My guess: they're isomorphic. My attempt: I ...
0
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0answers
13 views

If there exists a cyclic extension of degree $p$ of $K$, then there exists a cyclic extension of degree $p^n$.

If $char K=p \neq0 $, let $K_{p}=\{ u^p-u : u\in K\}$. Show that if there exists a cyclic extension of degree $p$ of $K$, then there exists a cyclic extension of degree $p^n$ for every $n \geq 1$. ...
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4answers
84 views

trying to understand what a polynomial ring is

I don' really understand what a polynomial ring is, maybe because the lack of examples. Consider for example the polynomial ring $\mathbb{Z}[x]$. Can you please tell me how this polynomial ring (its ...
0
votes
1answer
37 views

maximal ideal problem [duplicate]

I want to solve this problem, but I have no idea how I can start: If $K$ is a field, $(a_1,...,a_n) \in K^n,$ and $I$ the ideal $I=\langle x_1-a_1,...,x_n-a_n\rangle$, then how can we prove that ...
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1answer
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How to solve $(t +1) ^2=5$ in Sage?

I want to execute solve($(t+1)^2==5, t$) in Sage and using the Ring $\mathbb {Z}[X]/(x^2-2)$ so that $(t+1)^2$ evaluates to $2t+3$ and the solution of the solve is [t == 1]. My question is how to ...
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1answer
45 views

Automorphisms in $Z_n$

I know an Automorphism is a group G that is an isomorphism where $h:G \rightarrow G$, (G is being mapped to itself) and that Aut(G) is the set of all Automorphisms in G. I was wondering how I would ...
4
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2answers
78 views

Automorphism that is an Involution of a finite group

I am studying for a final and am trying to solve this problem: Let $G$ be a finite group with an automorphism $\sigma:G\rightarrow G$ such that $\sigma \circ \sigma=1$ and whose only fixed point is ...
3
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2answers
51 views

Find the number of subgroups in $Z_p \times Z_p \times Z_p$

let $p$ be a prime number ; I want to find the number of subgroups in $G = Z_p \times Z_p \times Z_p$ $(Z_p = \mathbb{Z}/p\mathbb{Z})$. I know that there is $p^2 + p + 1$ copies of $Z_p$ in $G$ for ...
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0answers
39 views

Show for $x\in G$ written as product of $hk$.

Show for $x\in G$ written as product of $hk$ for $h \in H$ and $k\in K$. Let $G$ be of order $p^km$ for $p$ is prime and does not divide $m$. $H=(x\in G\mid x^{p^k}=e)$ and $K=(x\mid x^m=e)$. G is not ...
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2answers
170 views

Artin Chapter 11, Exercise 9.12, polynomials without common zeroes [closed]

How do I show that the three polynomials $f_1 = t^2 + x^2 - 2$, $f_2 = tx - 1$, $f_3 = t^3 + 5tx^3 + 1$ generate the unit ideal in $\mathbb{C}[t, x]$? Artin mentions two approaches: by showing that ...
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2answers
65 views

On a proof that left artinian implies left noetherian

Questions: [Refer to below] Could one elaborate on $\rm\color{#c00}{(a)}$, $\rm\color{#c00}{(b)}$ and $\rm\color{#c00}{(c)}$ ? My thoughts : $\rm\color{#c00}{(a)}$ For $r+J\in R/J$ and ...
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3answers
39 views

orthogonal and special orthogonal group of dimension $2$, group of isometries of $S_1$, $\mathbb{R}^2$ [closed]

In my abstract algebra class, my teacher gave us this problem as to help review for the final. Unfortunately, I am not very well versed with linear algebra so I don't understand all that well what ...
4
votes
1answer
51 views

If a nilpotent group $A$ act on a group $G$ then $G$ is solvable.

Theorem: If a nilpotent group $A$ act on $G$ by automorphism and $C_G(A)=e$ then $G$ is solvable. I hope that the statement of theorem is true. It should belong to Hartley, but when I googled it I ...
4
votes
2answers
42 views

Why $P_1\neq P_1P_2$?

Question: If $P_1,P_2$ are distinct prime ideals of an artinian ring, why is it that $P_1\neq P_1P_2$? I know that prime ideals of an artinian ring are maximal, but still, I can't see why ...
3
votes
2answers
64 views

Show that $\mathbb Q(\sqrt p) \not\simeq\mathbb Q(\sqrt q)$

I'd like to show that for $p,q$ distinct primes, the extensions $\mathbb Q(\sqrt p),\mathbb Q(\sqrt q)$ are not isomorphic. I don't really have knowledge of the "high-level language" of algebraic ...