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

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To show that $I$ is an ideal of $\Bbb Z[\sqrt 2]$ and $I$ is a maximal ideal of $\Bbb Z[\sqrt 2]$.

Let $I = \{a +b\sqrt 2 \in \Bbb Z[\sqrt 2] \mid a$ and $b$ are both multiple of $5 \}$. To show that $I$ is an ideal of $\Bbb Z[\sqrt 2]$ and $I$ is a maximal ideal of $\Bbb Z[\sqrt 2]$. Then to find ...
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Show that the ideal generated by $4$ in $\mathbb Z_{12}$ is not a prime ideal.

Show that the ideal generated by $4$ in $\mathbb Z_{12}$ is not a prime ideal. Hint: Give a counter-example This is my rough proof to this question. I was wondering if anybody can look over it ...
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Determine U(R[x])

Determine $U(\mathbb{R}[x])$. What I have: If $x$ is a nonzero real number, then $x \cdot \frac{1}{x}$=1. So $x$ has a multiplicative inverse, and both $1$ and $-1$ belong to $x \in$ $U(\mathbb ...
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1answer
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Representing an algebraic number in an algebraic number field

Let $\alpha$ be an algebraic number (with minimal polynomial $f$). Then $\mathbb{Q}(\alpha)$ is an algebraic number field, and elements of $\mathbb{Q}(\alpha)$ can be represented as polynomials of ...
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Ideals in the ring of $n\times n$ complex matrices

I want to find the left and right ideals in the ring of $n\times n$ complex matrices. Let's start with the left ideals: A subset $I$ of $R$ is called a left ideal of $R$ if it is an additive ...
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27 views

To show that there exists $a_1 \in M_1$ and $a_2 \in M_2$ such that $a_1 + a_2 =1$.

Let $M_1$ and $M_2$ be two distinct maximal ideals of a ring $R$ then to show that there exists $a_1 \in M_1$ and $a_2 \in M_2$ such that $a_1 + a_2 =1$. Help Needed!
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Is the direct product $\mathbb{Z}_4 \times \mathbb{Z}_2$ cyclic?

Is the direct product $\mathbb{Z}_4 \times \mathbb{Z}_2$ cyclic?. How would you check this?
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Whether an embedding is an automorphism

Let $K/F$ be a field extension and let $\sigma$ be an embedding from $K$ into $K$ over $F$. If $K/F$ is algebraic, prove that $\sigma \in Aut(K)$. I know how to prove the case when $K/F$ is finite. ...
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Galois group of the simple extension

Let $K=Q(\sqrt3,\sqrt5)$. Show that the extension is $K/Q$ is simple and also Galois extension. Determine its Galois group. I showed that extension is simple because $K=Q(\sqrt{3}+\sqrt{5})$ But I ...
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Group theory and Complex Analysis

Actually this time I am having interest in group Theory and complex analysis. So I want to study the topics which relate both. So please suggest me which topic or book should I read to explore the ...
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A question about endomorphism rings of elliptic curves

This is probably a very trivial question, but I haven't been able to find a rigorous explanation anywhere so far or at least haven't understood it. Assume we have an elliptic curve $E$ over ...
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25 views

order of a subgroup of an alternating group

Let $U(2,4)$ be the union of all $2$-Sylow subgroups of Alternating group $A_4$. Also let $K(2,4)$ be the subgroup of $A_4$ generated by $U(2,4)$. Then what is the order of $K(2,4)$? I don't know how ...
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Product of principal ideals: $(a)\cdot (b) = (a b)$

In which kinds of rings $R$ does the following hold: $$(a)\cdot (b) = (ab) \; ?$$ With $a, b\in R$, $(a)$ denoting the (two-sided) ideal generated by $a$ and the multiplication of ideals $I, ...
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63 views

Number of distinct prime ideals

How many distinct prime ideals are there for $\mathbb Q[x]/(x^5 - 1)$? I've got $x^5 - 1$ as a reducible polynomial in $\mathbb Q[x]$. Also $x^5 - 1 =(x-1)(x^4 + x^3 +x^2 + x + 1)$. Here $(x^4 + ...
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1answer
31 views

Isomorphism of two non-abelian groups of order $pq$

Let $p$ and $q$ be two primes such that $q\mid p-1$. Suppose $\phi, \varphi$ are two non-trivial homomorphism from $\mathbb{Z}_q$ to $Aut(\mathbb{Z}_p)$. How to define an isomorphism from ...
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The splitting of an ideal

Let $K = \mathbb{Q}(\sqrt{-5})$. Now the ring of integers $\mathcal{O}_{K}$ is $\mathbb{Z}[i\sqrt{5}]$. I want to describe the ideal $(2)$ in $\mathbb{Z}[i\sqrt{5}]$ using the prime factorization. ...
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1answer
13 views

Occurrences of trivial representation is equal to dimension of $\{v\in V:\varphi(g)v=v\}$.

Suppose $\varphi\colon G\to GL(V)$ is a complex representation with character $\psi$. If $W=\{v\in V:\varphi(g)v=v,\ \forall g\in G\}$, why is $\dim W=(\psi,\chi_1)$, where $\chi_1$ is the principal ...
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Number of Sylow $p$-subgroups of a direct product of groups

Let $G$ be the group $S_4\times S_3$ . Prove or disprove the following: a $2-$Sylow subgroup of G is normal a $3-$Sylow subgroup of G is normal I've got $|S_4\times S_3|=144$ and the group as not ...
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(Theoretical Questions) Determine the following statements are true or false.

It is true or false answer so lets get started. False True False, In any ring R, a maximal ideal is an ideal M that is maximal in the set of all proper ideals of R, i.e. M is contained in exactly ...
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Prove or disprove $R= \mathbb Q[x]/\langle x^3-x^2+x-1 \rangle$ is an integral domain.

I've got $R$ is not a field since the polynomial is reducible in $ \mathbb Q[x]$. Is it possible to say anything from this?
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Example of a monomorphism and epimorphism that is not isomorphism. [duplicate]

I'm starting with a course of Introduction to Category Theory, and perhaps is dumb what I'm asking but I'm looking for an example of a monomorphism and epimorphism that is not isomorphism. Can you ...
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Infinite non-abelian $ p $-groups.

Is it true that every nilpotent group is a solvable group? It is true for finite $ p $-groups, but I am not sure about infinite $ p $-groups.
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a question about abstract algebra,the order of $\Bbb Z_{5}[x]/ (x^3+x+1)$

Firstly, I have proven that $x^3+x+1$ is irreducible in $\Bbb Z_{5}[x]$,then how can I know the order of $$\Bbb Z_{5}[x]/ (x^3+x+1),$$ where $(x^3+x+1)$ means the ideal generated by $x^3+x+1$. Can ...
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Is the condition of PID necessary?

In Gallian's Contemporary Abstract Algebra, one of the exercises is to show that if $D$ is a principal ideal domain, then show that every proper ideal of $D$ is contained in a maximal ideal of $D$. ...
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What's wrong with my example?

I have been asked to show that if $V\subset k^n$ is an affine algebraic variety over an algebraically closed field $k$, and $dom(f) = V$ for some $f\in k(V)$ then $f$ lies in $k[V]$. Here, $k(V)$ is ...
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How to show that $\sqrt[3]{p} \neq a+b\sqrt{q}$ for any $a,b\in \mathbb{Q}$?

p and q are primes, they could be same or different numbers. a and b are in $\mathbb{Q}$. I'm trying to prove that $\sqrt[3]{p} \notin \mathbb{Q}(\sqrt{q})$, and I think this is how to do it. After ...
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Is $R_1 \oplus 0$ a free $R_1 \oplus R_2$-module?

Suppose $R_1$ and $R_2$ are unital rings. Consider $R_1 \oplus \{0\}$ an $R_1 \oplus R_2$-module. Is this a free module? I am thinking it's not, since there are relations. How can I take this ...
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Infinitely many direct sum decompositions of $M$ into direct sum of irreducible $\mathbb{C}G$-modules?

I teaching myself character theory, but I don't understand a problem statement from Dummit and Foote, Exercise 18.3.4. Prove that if $N$ is any irreducible $\mathbb{C}G$-module, and $M=N\oplus N$, ...
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1answer
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Condition for module to not be free

Let $M$ be a module over a ring $R$ with $1$. Suppose that $rm=0$ for some nonzero $r \in R$ and $m \in M$. Can we conclude $M$ is not free? Here is my attempt. Suppose $M$ is free with basis ...
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1answer
15 views

Successive localizations of a module

I am trying to prove the following: Suppose that $p\subseteq q$ are prime ideals in $R$ and $M$ is an $R$-module. Then the the localization of the $R$-module $M_{q}$ at $p$ is $M_{p}$, i.e., ...
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32 views

Show $G/N $ is cyclic

if $G$ is cyclic, and $N$ is normal to $G$, then $G/N$ is cyclic Can anyone give me a git to start this question? Thanks
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25 views

Is the direct product $\Bbb Z \times \Bbb Z$ with operation $(n,m)+(p,q):=(n+p,m+q)$ a cyclic group?

Is the direct product $\Bbb Z \times \Bbb Z$ with operation $(n,m)+(p,q):=(n+p,m+q)$ a cyclic group? I know its not a cyclic group but how would i show this in a formal way?
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Irreducible Polynomial-Am I doing this wrong?

Ok,this problem might appear a bit trivial but I have some doubts..If it's not a burden take a look and comment! Let $F$ be a finite field of characteristic equal to $p$ and $ƒ(x)=x^p-α$ $∊F[x]$.Show ...
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Invariant point (flex) invariant under projective transformations.

Let $C$ be a projective curve in $\mathbb{P}_2$ defined by a homogeneous polynomial $P(x, y, z)$ and let $\alpha$ be a linear transformation of $\mathbb{C}^3$. Let $Q$ be the homogeneous polynomial $Q ...
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Splitting field of $f(x)=x^4+3$ in $\mathbb{Q}[x]$

I am trying to find the splitting field of $f(x)=x^4+3$ over $\mathbb Q$. It is irreducible, and the roots are ...
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Why do we not lose any generality by proving it only for finitely generated groups.

In the proof of following theorem, in a paper by Farkas- Here $\Delta(G) = \{ g \in G : |G:C_G(g)| < \infty \}$ and $U_1(\mathbb{Z}G) $ is the set of normalized units of the integral group ring ...
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Finding an ideal such that $\mathbb{Z}[x]/I \cong \mathbb{Z}[i]$.

On this released exam, it asks at 2g (slightly modified wording): Give a brief example or show there does not exist an ideal $I$, $I \subseteq \mathbb{Z}[x]$ such that $\mathbb{Z}[x]/I$ is ...
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2answers
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Image of a normal subgroup under automorphism is the normal subgroup

Let $G$ be a finite group. Let $H$ be a normal subgroup of $G$ such that the order of H and the index of $H$ in $G$ are relatively prime. Let $f$ be an automorphism of G and let $J = f(H)$. Prove ...
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Is a polynomial ring over a UFD in countably many variables a UFD?

Let $R$ be a UFD. It is well know that $R[x]$ is also a UFD, and so then is $R[x_1,x_2,\cdots,x_n]$ is a UFD for any finite number of variables. Is $R[x_1,x_2,\cdots,x_n,\cdots]$ in countably many ...
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Are complex split-octonions isomorphic to a more easily-defined algebra?

I write fiction and nonfiction, both which are mathy. My fiction is not usually supermathy but I'm working on a fictional story that has some math in it, and I prefer accuracy to mathbabble. I'm ...
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Finding The Number Of Inversions In A Permutation

Let the be the following permutation: $(1 5 4)(3 6)\in S_6$ How do I count the number of inversions to calculate the sign of the permutation? $(1 5 4)(3 6)=(1 5)(1 4),(3 6)=3$ so it has an ...
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For a polynomial $f\in K[x]$, when is there a constant $c\in K$ such that $f+c$ is irreducible?

I was working on a different problem when the following question occurred to me: For a polynomial $f\in K[x]$, is there always a constant $c\in K$ such that $f+c$ is irreducible? Obviously this is ...
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Factoring polynomials over finite fields

I'm having some trouble understand how to test for irreducibilty of polynomials over finite fields. For example, in my text book I see $5x^4-2x^3 +9x -1$ is irreducible in $\mathbb{Q}$. They then go ...
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How to show that the following ideal is prime/maximal?

If $I$ is the set of polynomials that can be written as $2p(x)+(x^2+x+1)q(x)$, how can I show that it is or isn't a prime ideal of $R=\mathbb{Z}[x]$? $I$ know that if $I$ consider two polynomials A ...
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Check the order of a subgroup of the alternating group $A_n$

Question : For a positive integer $n \geq 4$ and a prime $ p \leq n$. Let $U_{p,n}$ denote the union of all p- syllow subgroups of the alternating Group $A_n$ on n letters . Also let $K_{p,n}$ denote ...
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Largest common divisor

Show that each common divisor $c_1 , \ldots , c_n \in \mathbb{Z}$ divides their largest common divisor. Use subgroups of the group $ \mathbb{Z}$. Could somebody help me? Please
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Ideal of Ring of holomorphic functions

Can you tell me a non trivial ideal of ring of holomorphic functions from C to C.
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1answer
35 views

Galois group of the extension $E:= \mathbb{Q}(i, \sqrt{2}, \sqrt{3}, \sqrt[4]{2})$

In order to make a smaller example for my question Galois group of the field of all constructible complex numbers, I am posing this new question. I know already, that E is a galois extension of ...
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1answer
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Squares in a Finite Field

Show that in any finite field,each of its elements can be written as the sum of two squares. Well,I hate to admit-this being also my first post-that I have not proven it yet.I tried to work on the ...
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4answers
103 views

If G is a group not cyclic then its order can be:

If G is a group not cyclic then its order can be: a)15 b)35 c)77 d)120 e)2011 Well, i know that if G is not cyclic then it is not isomorphic to Zn, but i think it does not help much. Any ...