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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How to factor polynomials in $\mathbb{Z}_n$?

How to factor a certain polynomial over $Zn$. for example factor the following polynomial into irreducible polynomials in $Z5$: $X^3+X^2+X-1$ or factor the following polynomial into irreducible ...
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1answer
21 views

Prove algebraic closure

Let $L/K$ be a field extension such that $L$ is algebraically closed. Show that $\{a\in L\mid[K(a):K]\lt\infty\}$ defines an algebraic closure of $K$. So this is the set of minimal polynomials ...
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Ramification group of valuations - need terminology

I am lost and need some terminology (also hopefully sources). Let $L/K$ be a Galois extension, and $w$ be a valuation of a $L$, lying above a valuation $v$ of $K$. Notice that I do not suppose that ...
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Please show that $W$ is a normal subspace of $\sigma$. [on hold]

Suppose that $\sigma$ is a symmetry transformation, $V$ is a space, and $W$ is a subspace of $V$. Please show that $W$ is a normal subspace of $\sigma$.
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52 views

Solution verification: proving that $2$ is not prime in $\mathbb{Z}[\sqrt{-3}]$

I just took my final exam for abstract algebra and have this problem stuck in my head. Prove that $2$ is irreducible in $\mathbb{Z}[\sqrt{-3}]$ but not prime. My Solution: Proving that it is ...
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1answer
13 views

Prove center of a group is a subgroup using one-step subgroup test

I'm not sure if this is correct. It doesn't seem so. If $a,b \in C$, then we must show $ab^{-1} \in C$. $$ab^{-1}x=axb^{-1}=xab^{-1}$$ This doesn't seem correct. I've seen two-step subgroup tests ...
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19 views

Given transpositions, can you express the permutation in matrix form?

I know that if you are given the matrix itself or the disjoint cycles, you can easily express the permutation as a product of transpositions, but if you are only given the transpositions, can you go ...
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Galois group of the splitting field of the polynomial $x^5 - 2$ over $\mathbb Q$

We know that the splitting field of $x^5 - 2 $ over $\mathbb Q$ is $\mathbb Q(2^{1/5}, \rho)$, where $\rho$ is a fifth root of unity. Therefore, $\left[\mathbb Q(2^{1/5} , \rho) : \mathbb Q \right] = ...
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1answer
34 views

How do i find maximum order of element in $S_{10}$ Group

Question is to find maximum possible order of an element in $S_{10}$ Group . Please someone help me through this .How to think of this intuitively .Thanks
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3answers
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Confused about a solution: proving every prime ideal is maximal

I'm looking at this solution to this problem: I'm getting thrown off by the special case where $n = 2$. If $n = 2$, why must it be that $x = 1$? All that we then know is that $x^2 = 1$ or that $x = ...
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2answers
42 views

Show that polynomial is reducible

Show that $p(x)$ = $x^3 + 3x^2 + 2x + 4$ is reducible in $\mathbb{Z}$$/$$7$$\mathbb{Z}$ Is the approach for this to factor it and then find a root? I'm a little confused on how to start. Any tips ...
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For any $p,q\in\mathbb{Z}[i]$, $\mathrm{N}(\gcd(p, q))$ must divide $\gcd(\mathrm{N}(p), \mathrm{N}(q))$

I'm studying for my final exam (abstract algebra) and am looking at an example where our professor was trying to compute the GCD of two elements of $\mathbb{Z}[i]$. Rather than directly applying the ...
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1answer
23 views

Show that $H$ is transitive on the set $G$.

Let $G$ be a group and let a be a fixed element of $G$. The map $\lambda_{a}: G \to G$, given by $\lambda_{a}(g) = ag$ for all $g \in G$, is a permutation of the set $G$. Note $H = \{\lambda_{a} ...
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32 views

Irreducible elements and unique factorization domain

Let $P=\{\frac{a}{3^n} : a \in \mathbb{Z}, n \in \mathbb{N}\}$. a) Which elements are irreducible in $P$: 4, 5, 6, 9, 10, 15? b) Find out, which one of rings: $ P$, $\mathbb{Z}[i\sqrt{5}]$, $P[x]$ ...
2
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1answer
73 views

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 ...
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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) $0_s \in f(I)$ ...
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1answer
64 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?
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1answer
26 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 ...
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1answer
104 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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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 ...
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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 ...
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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 ...
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20 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 ...
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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?
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2answers
59 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 ...
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33 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 ...
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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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$K \unlhd H\unlhd G$, 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 ...
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1answer
24 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 ...
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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 $G$. ...
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1answer
30 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 ...
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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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Relation between generators of a free graded $k[x,y]$-module and a free graded submodule

Let $M = \bigoplus_{i=1\ldots 5}R(m_i)$ be a free $\mathbb{Z}^2$-graded $R$-module where $R=\mathbb{Z}_p[x,y]$ and $N=\bigoplus_{i=1\ldots 5}R(n_i)$ a free graded submodule of $M$. Define the ...
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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) ...
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1answer
21 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
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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
46 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
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1answer
27 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, ...
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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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37 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 ...
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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 ...
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1answer
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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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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
14 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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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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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
36 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 ...
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3answers
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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 ...