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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Question about a passage in Fulton and Harris

So I was reading the first chapter of Fulton and Harris and they are determining the representations of $S_3$. I came along this passage and had some questions What do they mean when they say "the ...
1
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1answer
51 views

Factoring $x^{15}−1$ into irreducible polynomials over $\mathrm{GF}(2)$

Factorize $x^{15}−1$ into irreducible polynomials over $\mathrm{GF}(2)$ The answer is $$(x+1)(x^2+x+1)(x^4+x+1)(x^4+x^3+1)(x^4+x^3+x^2+x+1)$$ but how would I find this? Please help.
-1
votes
1answer
22 views

What is the name for this $R$-module?

If $M$ is a $R$-module such that for all $x,y$ in $R$ and $m$ in $M$ then $x.y.m=0$ ... then: how is called $M$?
4
votes
0answers
37 views

Show that $t^n-1 \mid t^m-1 \Leftrightarrow n\mid m$ [duplicate]

I want to prove the following lemma: $t^n-1$ divides $t^m-1$ in $F[t, t^{-1}]$ if and only if $n$ divides $m$ in $\mathbb{Z}$. I have done the following: $\Leftarrow $ : $n\mid m \Rightarrow ...
4
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0answers
32 views

Splitting even degree polynomials

I have an octic equation (degree $8$) and a sextic equation (degree $6$) in $\Bbb Z[x]$ with very large coefficients (size several hundred bits) that I know splits into two quartics and two cubics ...
1
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1answer
94 views

Solve $x^3+x+3=0$ by Galois's theory

Solve with radicals the following equation $x^3+x+3=0$, using Galois Theory. I'm just starting learning this and I do not have many ideas.
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votes
2answers
33 views

Ring homomorphism and ideal that contains the kernel [on hold]

If $f:R\rightarrow S$ is a ring homomorphism and $I$ ia an ideal of $R$ such that $ker(f) \subseteq I$ then $f^{-1}(f(I))=I$ We know that $I\subseteq f^{-1}(f(I))$ but how can I use that $ker(f) ...
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1answer
36 views

A symmetric group question. [on hold]

Determine the integers $n$ such that there is a Surjective homomorphism from the symmetric group $S_n$ to $S_{n-1}$. It is a question from Artin's book. Exercise 7.5.8
3
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2answers
54 views

Kernel of ring homomorphism

Let $\phi: R \to R'$ be a ring isomorphism and $I$ an ideal of $R$. Define $\phi(I)=\{\phi(i): i \in I\}$. Show that $\frac RI \cong \frac {R'}{\phi(I)}$. To use the first isomorphism theorem, ...
1
vote
1answer
40 views

For each $n >2$ find all $m \in \mathbb N$ such that there exist a group $G$ of order $m$ and a surjective homomorphism $f: A_n \rightarrow G$

For each $n >2$ find all $m \in \mathbb N$ such that there exist a group $G$ of order $m$ and a surjective homomorphism $f: A_n \rightarrow G$< I tried to use the theorem of isomorphism, and a ...
4
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1answer
35 views

A $\mathbb Z/p\mathbb Z[G]$ submodule with no complement

Let $G$ be a group acting on a set $X$ of size $n$. Suppose $G$ acts doubly transitively. If $p$ is a prime, this naturally gives a permutation representation on the vector space over $\mathbb ...
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2answers
51 views

If $G = S_5$ and $H = \{g \in G \mid g^{5} = e\}$ how could I determine and prove whether or not $H$ is a subgroup of $G$?

I think that the this group contains the 5 element cycles and the identity e but overall I'm not sure how to prove that the product of the 2 members of H is also a 5 cycle or e.
3
votes
3answers
67 views

Why does $\sqrt{6} + \sqrt{10} + \sqrt{15}$ have four conjugates?

I am having trouble understanding how algebraic number $\sqrt{6} + \sqrt{10} + \sqrt{15}$ has four conjugates. Minimal polynomial is $x^4-62 x^2-240 x-239$ according to Wolfram Alpha. Factorized: ...
13
votes
1answer
75 views

Is the ring of holomorphic functions on $S^1$ Noetherian?

Let $S^1={\{ z \in \Bbb{C} : |z|=1 \}}$ be the unit circle. Let $R= \mathcal{H}(S^1)$ be the ring of holomorphic functions on $S^1$, i.e. the ring of functions $f: S^1 \longrightarrow \Bbb{C}$ which ...
2
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1answer
31 views

Showing a nonabelian group of order 21 has an automorphism that is not inner.

I've seen at least 3 proofs of this on here, but most involved techniques I don't think I'm comfortable with, so I wanted to see if this one works: Since $21=3\cdot 7$, up to isomorphism there's only ...
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0answers
27 views

Groups,subgroups, normal subgroups [on hold]

Let $G$ be the group of all $2\times 2$ real matrices $\left( \begin{array}{ccc} a & b \\ 0 & d \end{array} \right) $ under matrix multiplication where $a,d\neq 0$. If $N=\left\{ \left( ...
0
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0answers
36 views

A problem about isomorphism in module theory

For a sequence of $R$-modules like this:$A \overset{f}{\rightarrow} B \overset{g}{\rightarrow}C$ such that $\mathrm{Im} f \subset\ker g$ and $B/\mathrm{Im}f \cong B/\ker g$. Then $\mathrm{Im}f=\ker ...
1
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1answer
43 views

Prove that a morphism $\alpha$ of $Fun(\mathcal{A},\mathcal{B})$ is an isomorphism iff each component $\alpha_A$, is an isomorphism in $\mathcal{B}$

I'm a computational engineer starting with a course of Introduction to Category Theory, and perhaps is extremely basic what I'm asking but I'm trying to learn how to make proofs in category theory ...
1
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1answer
30 views

Properties of isomorphism in module theory [on hold]

I have two exercises, but I can't solve them: a. If $X, A, B $ are $R$-modules with $A \subset B \subset X $, prove that if $X/A \cong X/B$ then $A=B$ b. If $X/A \cong X$ then $A=0$
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0answers
34 views

Inducing representations from a subgroup of finite index.

Let $G$ be a group and $H$ a subgroup of finite index. Let $(\sigma , W) $ be a irreducible representation of $H$ (which need not be finite dimensional). 1) When can we extend this representation of ...
1
vote
1answer
26 views

I have to show no proper intermediate fields exist between $Z_2$ and $GF(2^3)$ [duplicate]

I have to show no proper intermediate fields exist between $Z_2$ and its overfield $GF(2^3)$, Can any one help?
3
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1answer
32 views

Prove the theorem of ideal (about g.c.d)

If $p_1,\ldots,p_n$ are polynomials over a field $F$, not all of which are $0$, there is a unique monic polynomial $d$ in $F[x]$ such that (a) $d$ is in the ideal generated by $p_1, \ldots, ...
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0answers
33 views

Irreducible component of a scheme over a dvr

Let $\mathcal M$ be a (reduced) quasi-projective scheme over a dvr (of mixed caracteristics), $R$. Suppose that the generic fiber $\mathcal M_{\eta_R}$ is (nonempty) smooth and irreducible of ...
0
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2answers
36 views

How do I construct a nontrivial linear representation of the group $G$=S' in $R^3$

$S'$ is defined as the unit circle. The product is defined as the sum of angles. How do I construct a linear representation in general? I don't know how to begin with this problem. Some ...
0
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0answers
44 views

Number of all possible groups of given order [duplicate]

Suppose $n=18$, then all possible groups of order $18$ is $5$. Among them $2$ are abelian and $3$ are non-abelian. Let $n$ be a natural number. How can I determine the number of all possible ...
1
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1answer
24 views

Difference between torsion and zero divisor

I'm not understanding what the difference is between a zero divisor and a torsion element of a module. My best guess is that the torsion elements are "vectors" and zero divisors are scalars. This ...
0
votes
1answer
38 views

If $gcd(a,b)=1$, then there exists integers x and y such that $xa + yb = 1$

Did not find this from this website... If $$ gcd(a,b)=1,$$ then there exists integers x and y such that $$xa+yb=1.$$ Now, the tip is to use particular corollary, that states: The class $[m]_{n}$ ...
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1answer
13 views

Show that there exists $\mathbb{Z}[i]$ modules $M$ and $N$ such that both have $13$ elements but are not isomorphic

$\textbf{Question:}$ Show the existence of two $\mathbb{Z}[i]$ modules $M$, $N$ such that both $M$ and $N$ have $13$ elements bot $M \not \cong N$ as $\mathbb{Z}[i]$ modules. $\textbf{My Attempt:}$ ...
4
votes
1answer
66 views

Is $(x)\otimes_{k[x]/(x^2)}(x)$ zero?

I am trying to decide if $(x)\otimes_{k[x]/(x^2)}(x)$ is zero. So I considered $x \otimes x$ which I rewrote as $1 \otimes x^2 = 1 \otimes 0 = 0$. But then I realized that $1$ does not live in ...
0
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2answers
63 views

Automorphism of $D_6$

I need find an explicit way to express the group $Aut(D_6)$, and I have not idea how write this group, maybe this is an semidirec product of some groups but I don´t see this. thanks.
2
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1answer
26 views

Why does $\operatorname{Ad}_h((S\otimes 1)(Q))=\epsilon(h)(S\otimes 1)(Q)$ in a quasi-triangular Hopf algebra?

I'm reading a proof that in a quasi-triangular Hopf algebra $H$, $(S\otimes 1)Q$ is $\operatorname{Ad}$-invariant. Here $Q=\tau(R)R$, where $R$ is the invertible element in $H\otimes H$ satisfying all ...
3
votes
1answer
43 views

Tangent space of quotient space

Let $\pi : M \rightarrow M/G$ be the canonical projection, where $M$ is a manifold and $M/G$ is a quotient manifold. Now, what can we say about $d \pi (p) : T_pM \rightarrow T_p(M/G)$? From my ...
2
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0answers
63 views

Fields of Research in Algebra [on hold]

I'm a last-year student in mathematics and I'm looking for a master degree in algebra. So I'm trying to understand what are the most interesting fields of research in algebra all around the world. ...
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1answer
26 views

Lie group and stabilizer quotient

Let $G$ be a Lie group and $$G_x:=\{g \in G; Ad^*_g(x)=x\}$$ the stabilizer, where $Ad_g^*$ is the adjoint of the adjoint representation. Now, I was wondering why $G/G_x$ has a manifold structure. ...
3
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1answer
42 views

What do we know about fields possessing an involution?

The field $\mathbb{C}$ of complex numbers has an involution, and the same is true of the field of algebraic numbers (the algebraic closure of $\mathbb{Q}$ as a subfield of $\mathbb{C}$) and of the ...
3
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1answer
33 views

Global Dimension of a Ring and its Localizations

Why is the following true? The global dimension of a noetherian ring $A$ is the supremum of the global dimension at its localizations at its maximal ideals: ...
7
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62 views

index 2 subgroups of the infinite product of Z/2Z

Is it possible to describe all the index 2 subgroups of the group $G = \prod_{i\in \mathbb{N}}\; \mathbb{Z}/2\mathbb{Z}$? For example, one can take the kernel of the $i$-th projection map ...
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0answers
31 views

Open subscheme in special fiber

Let $X$ be a projective scheme over $R$ a discrete valuation ring with generic fiber irreducible. Can an open subscheme of $X$ be contained in the special fiber of $X$ ? Or is it true that every open ...
2
votes
2answers
47 views

Permutations: Interpreting Image Notation

I have a problem in interpreting permutation. I think the definition and my interpretation of it don't match each other. Let $\sigma=(1\ 2\ 4\ 3)$, and $\tau=(1\ 3\ 2\ 4)$ in one-line notation. I ...
3
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1answer
54 views

proving that the symmetric group $S_x$ is not finitely generated where $x$ is infinite [on hold]

it seems pretty trivial, but I have trouble of showing it. I also wonder of good approach of proving that a group is not finitely generated.
2
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1answer
14 views

Commutative Monoid - matrix set

Let $M$={$\begin{bmatrix} a & b & c \\ c & a & b \\ b & c & a \end{bmatrix}|a,b,c\in \mathbb{R}, a+b+c=0$}. The matrices in $M$ are a special kind of Toeplitz matrices ...
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0answers
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how is the formula for solving cubic equation derived directly from sum of roots, product of roots and sum of product of roots taken 2 at a time?

In $ax + b = 0$, let the root be $p$. We know $ p = -b/a$ In $ax^2 + bx + c = 0$, let the roots be $p$ and $q$. We know, $$\frac{-b}{a} = p+q $$ $$\frac{c}{a} = pq $$ Now, ...
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0answers
39 views

Proving that $t^{p^r}-a$ is irreducible when $a\in k$ is not a $p$th power

Let $p$ be an odd prime, $F$ a field of characteristic $0$ and $a\in F$ with $a\neq 0$. Assume $a$ is not a $p$th power in $F$. Prove that for every positive integer $r$, $t^{p^r}-a$ is irreducible ...
3
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1answer
39 views

Is it true that every prime ideal of height one is principal? [on hold]

Is it true that every prime ideal of height one is principal ? Please help
3
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2answers
51 views

$R$ a local ring, also a PID. $I,J$ ideals from $R$. Show that $I \subseteq J$ or $J \subseteq I$

$R$ a local ring, also a PID. $I,J$ ideals from $R$. Show that $I \subseteq J$ or $J \subseteq I$ My brief attempt to try use Bezout theorem at a PID. but unsuccess.. Thanks any help.
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1answer
75 views

Is there an infinite topological meadow with non-trivial topology?

For reference meadows are a generalization of fields that were designed to be compatible with the requirements of universal algebra. Specifically a meadow is a commutative ring equiped with an ...
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79 views

Comments about “Topics in Algebra” by I.N. Herstein and “Abstract Algebra” by Dummit/Foote? [on hold]

Today, I got two gifts from my research mentor: "Topics in Algebra" by I.N. Herstein and "Abstract Algebra" by Dummit/Foote. I am very happy and grateful for his gifts, but I already have been ...
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1answer
32 views

The difference between Quotient Set and other definition

it a new course and material that I learn, we defined 2 pretty similiar definitions and I didnt understand what is the difference between the definitions. Definition 1: A subset $T\subseteq X$ is ...
4
votes
2answers
53 views

Finding the kernel of maps between (polynomial) rings

If I have a map between rings like $f\colon k[x_1,x_2]\to k[t],x_1\mapsto t^2-1,x_2\mapsto t^3-t$, how can I prove that the kernel is $\mathfrak{a}=(x_2^2-x_1^2(x_1+1))$? I see that ...
2
votes
2answers
24 views

Let $H$ be normal subgroup of $G$. If $G/H$ is cyclic group generated by $aH$, prove that $G=KH$ where $K=\langle a\rangle$.

Let $H$ be normal subgroup of $G$. If $G/H$ is cyclic group generated by $aH$, prove that $G=KH$ where $K=\langle a\rangle $. I would like someone to check my solution. First of i will prove that $G$ ...