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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Doubt related to quotients (group or ring)

I was reading some notes about ring theory and modules and I've encountered with the following isomorphism: $\mathbb (R[X]/ \langle x^3-1\rangle)/ \langle x-1\rangle \cong \mathbb R[X]/ \langle x-1 ...
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Surjections from free groups

I am stuck on the following: How do i go about finding surjections from the free group of rank 2 $\mathbb{F}_2$ = $\mathbb{ Z∗Z}$, to the finite group of two element $\mathbb{Z}_2$. Also,how would ...
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18 views

infinite version of lagrange theorem

Let G be a group and H and K subgroups of G. If K ⊂ H ⊂ G and K has finite index in G, then prove [G : K] = [G : H][H : K]. Obviously if we know G is finite, then we are done by Lagrange Theorem. ...
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1answer
36 views

The order of element in $\mathbb{Z} / 2^{2014}\mathbb{Z}$

Find the smallest integer $n$ such that $2^{2014}|17^n-1$. i.e. Find the order of $17$ in $(\mathbb{Z}/ 2^{2014} \mathbb{Z})^{\times}$. I think we have to use the lifting the exponent lemma: If ...
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1answer
50 views

There is no homomorphism from $\mathbb{Z_8} \times \mathbb{Z_2} \times \mathbb{Z_2}$ onto $\mathbb{Z_4} \times \mathbb{Z_4}$

If such a homomorphism $\phi$ existed, then the first isomorphism theorem says that $|\ker \phi| = 2$. Since $\mathbb{Z_8} \times \mathbb{Z_2} \times \mathbb{Z_2}$ is abelian, then every subgroup is ...
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1answer
21 views

Galois field extension and number of intermediate fields

Given the galois extension $L/K$, where $L=\mathbb{Q}(\zeta, \sqrt[3]{2})$. I've already calculated that $[L:\mathbb{Q}]=6$. I'm trying to prove that the number of intermediate fields is also 6, ...
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Let $K = \mathbb{Z_p(t)}$ be the field of rational functions and let $f(x) = x^p - x - t \in K[x]$ and let $E/K$ be the splitting field of $f(x)$.

Show $f(x)$ is not solvable by radicals and $Gal(E/K) \cong Z_p$. I was just reading Galois theory from the textbook "Galois Theory - Rotman" and I stumbled acrossed an exercise that looked ...
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1answer
32 views

Number of elements of a prime ideal's coset

In $\mathbb{Z}[x]$, let $$I = \{ f(x) \in \mathbb{Z}[x]\mid f(0)\text{ is an even integer}\}$$ Prove that $I= \langle x, 2\rangle$ Is $I$ a prime ideal of $Z[x]$? Is $I$ a maximal ideal? How many ...
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37 views

If $n|m$ prove that the natural surjective projection $\pi: \mathbb{Z_m} \rightarrow \mathbb{Z_n}$ is also surjective in units

Not sure if this is the right proof: since $n|m \Rightarrow n \leq m$, then if we factor $m = p_1^{\alpha_1}p_2^{\alpha_2}\ldots p_k^{\alpha_k}$, then $n = p_1^{\beta_1}p_2^{\beta_2}\ldots ...
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2answers
25 views

Intersection of all $p$-Sylow subgroups is normal

Let $G$ be a finite group, $p$ a prime number that divides $|G|$ and $O_p(G)=\bigcap_{P \in Syl_p(G)}P$. Prove that 1) $O_p(G) \lhd G$ 2) $O_p(G)$ is maximal among the normal $p$-subgroups of $G$. ...
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24 views

Definition of $\sum_{a \in A} I_a$

If $(I_a)_{a \in A}$ a family of ideal of $K[x_1,x_2, \dots, x_n]$, I have the following definition in my notes: $$\sum_{a \in A} I_a=\{ a_{i1}+a_{i2}+ \dots+ a_{ij} | a_{ij} \in I_{a_j} \}$$ Is ...
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19 views

A reduction of Cayley-Hamilton to the complex case [on hold]

I found an interesting proof of the general Cayley-Hamilton theorem but i didn't quite get all of it. I'm trying to make sense of this reduction argument. I'm not that familiar with algebras so this ...
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2answers
24 views

Is there a commutative ring $R$ with an idempotent endomorphism $f$ that cannot be expressed as $f(x)=sx$ for some idempotent $s \in R$?

For any element $s$ of a commutative monoid $M$, the following are equivalent. idempotency (that is, $s^2=s$). self-distributivity (that is, $s(xy) = (sx)(sy)$ for all $x,y \in M$). the function $M ...
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2answers
122 views

Algebraic operations with tensor products and direct sums.

When first learning about tensor products and direct sums of vector spaces, the analogy of multiplication and addition of vector spaces is sometimes used to help with the intuition. If we look at the ...
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1answer
36 views

Finding a polynomial satisfying the equation

For $$ f: x^6+3x^4-4 \\ g: x^5-x^4+5x^3-5x^2+6x-6 $$ how do I find a polynomial $a \in \mathbb{Q}[x]_{(\deg f-\deg \gcd(f,g))}$ so that a polynomial $b \in \mathbb{Q}[x]$ exists when ...
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1answer
28 views

How to find the splitting field?

How can I find the splitting field of $x^n+1 \in \mathbb{Q}[x]$ ?? If we have for example, $x^n-1 \in \mathbb{Q}[x]$ we would do the following: $$x^n-1=(x-1)(x^{n-1}+x^{n-2}+ \dots +x+1)$$ So, the ...
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2answers
25 views

If H and K are finite subgroups of G (another proof )

I have a question and it's solution , but I want another proof if there exist . Thanks
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21 views

Quick algebra question regarding fields and their elements

I'm not sure if I'm getting this right. Assume we have a field $F$ and its four elements are $a,b,c,d$ and have $a+b=ab=c$. First of all, where is my mistake in the following? \begin{align} ...
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0answers
36 views

Does the five lemma hold true for Lie algebras?

According to wikipedia, the Five Lemma is true in Abelian categories. But the category of Lie algebras is not Abelian. Then is the Five Lemma still true for Lie Algebras?
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2answers
21 views

Computing the inverse of an element in $\mathbb{Z}_5[x] / \langle x^2+x+2\rangle$

How does one calculate the inverse of $(2x+3)+I$ in $\mathbb{Z}_5[x] / \langle x^2+x+2\rangle$? Give me some hint to solve this problem. Thanks in advance.
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13 views

Definition of Linear independence of algebraic $1$-forms

"Suppose that $a,b,c,d$ are linearly independent algebraic $1$-forms on $\mathbb{R}^n$". What does it mean for algebraic $1$-forms to be linearly independent? I have looked through my notes and ...
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17 views

if group $G$ has three elements $a$,$b$,$c$ with the property that the order of each two of them is coprime,and $c=ab$ then $G$ is not solvable.

if group $G$ has three elements $a$,$b$,$c$ with the property that the order of each two of them is coprime,and $c=ab$ then $G$ is not solvable. this question is about my last question posted which ...
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31 views

Irreducible representations of $S_n$ [duplicate]

I want to prove that $S_n$ has an irreducible representation of dimension $n-1$. Intuitively, I know that the $\forall n$, the trivial representation is irreducible, and there should be some ...
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1answer
46 views

How to find the basis?

So if there is a n-dimensional space (x_1, x_2, ..., x_n) over a field, How can I find a basis which together with (c_1, c_2, ..., c_n) (where all c_i's are non-zero constants from this field) forms a ...
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1answer
48 views

Prove that every group $G$ whose order is the form $|G|=p_1^{\alpha_1}p_2^{\alpha_2}p_3^{\alpha_3}$ is not solvable [on hold]

Prove that every group $G$, whose order is the form $|G|=p_1^{\alpha_1}p_2^{\alpha_2}p_3^{\alpha_3}$, where $p,q,r$ are distinct prime numbers and $\alpha_i >1$, is not solvable. Any hint or ...
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1answer
57 views

$G/Z(G) \cong \mathbb Z_p \times \mathbb Z_p$ then $p||Z(G)|$

Problem Let $G$ be a finite group with $G/Z(G) \cong \mathbb Z_{p} \times \mathbb Z_{p}$. Then $p| |Z(G)|$. My attempt at a solution Consider the action of $G$ on itself by conjugation. By the ...
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45 views

Permutation group of a set

If we let $G$ be a finite permutation group of a finite set $X$ and assume that $G$ has exactly 2 orbits of the same cardinality, how can we show that there is some permutation in G that has no cycles ...
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0answers
15 views

Homotopy of morphisms of double complexes in Cartan Eilenberg

Let $s\colon f_1\cong f_2$ be a homotopy of two morphisms of complexes $A\to A'$ and $t\colon g_1\cong g_2$ be a homotopy of two morphisms of complexes $C\to C'$. I want to understand why ...
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1answer
113 views

algebra with topology homework problem

Hello Everyone, I have this homework problem, I'm going to share what i have so far, not sure if Im in the right path. First, I have: $$f \sim g \, \Leftrightarrow \,x_0 \in \mathbb{R^n}, \exists ...
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1answer
38 views

Simple questions about the Jacobson Radical

Questions: [See below] $\rm\color{#c00}{(1)}$ What structure has $R/I$ when $I$ is a maximal left ideal? I know that if $I$ is a bilateral ideal then $R/I$ is a ring and, moreover, if $I$ is ...
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2answers
42 views

Which are the nine Sylow $2$-subgroups of $S_3 \times S_3$? What is the only Sylow $3$-subgroup of $S_3 \times S_3$? And the most important…Why? [on hold]

Which are the nine Sylow 2-subgroups of $S_3 \times S_3$? What is the only Sylow 3-subgroup of $S_3 \times S_3$? And the most important... why? I am doing an independent study of Abstract Algebra. ...
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1answer
31 views

Field at which $f(x)$ splits [on hold]

Let $f(x) \in \mathbb{Z}_p[x]$. Show that there is a finite field $\mathbb{F}_{p^n}$ at which $f(x)$ splits. And if $f(x)$ is also separable, show that $f(x) \mid x^{p^n}-x$. Could you give me some ...
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1answer
35 views

The union of finite field extensions is a finite field extension

Assume that all elements under discussion are algebraic over $F$. Let the notation "$K=F(A)$" mean that $A\subseteq K$ and there is an injective homomorphism $\sigma:F\to K$, and every element of $K$ ...
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Extensions of $\mathbb{Z}$ by $\mathbb{Z}_2$

Well. My question is very concrete. Does anybody know all the groups $G$ such that it fits in a short exact sequence $1\to \mathbb{Z}\to G\to \mathbb{Z}_2 \to 1$, where $\mathbb{Z}_2$ are the integers ...
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1answer
28 views

Which are the generators of $S_3 \times S_3$ (that has 36 elements)? Please do not confuse with $S_3$ that has 6 elements. [on hold]

Which are the generators of $S_3 \times S_3$ (that has 36 elements)? Please do not confuse with $S_3$ that has 6 elements. I am doing an independent study of Abstract Algebra. Your help is very ...
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1answer
46 views

Which is the identity element of $S_3 \times S_3$ (that has 36 elements)?

Which is the identity element of $S_3 \times S_3$ (that has 36 elements)? (Please do not confuse with $S_3$ that has 6 elements.)
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4answers
118 views

Roots of different irreducible polynomials are algebraically independent

Let $F$ be a field, and let $f$ be a monic irreducible polynomial over $F$. Let $\alpha$ be a root of some other monic irreducible $g\ne f$. Then is $f$ still irreducible in $F(\alpha)$? Is it true ...
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1answer
25 views

Stuck on Finding Roots in a Field

So I'm working in the field $F_{5}[x]/(x^2+2)$, and I have to find the roots of $t^2+2$ here. Before I start, note that when I write = it's in this field. So I know that the roots have to be less ...
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2answers
66 views

I do not understand this Abstract Algebra notation. Can you help me?

$$ \langle((a,b),1),(1,(c,d))\rangle $$ I am doing an independent study of Abstract Algebra. I found the notation that is in the attachment in a Standford webpage, but I do not understand the meaning. ...
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20 views

Nice Proof $\mathbb{Z}[\sqrt{6}]$ is a euclidian domain wrt absolute norm map

I know that $\mathbb{Z}\left[\sqrt{6}\,\right]$ is a Euclidian domain with respect to the absolute valued norm map $x+y\sqrt{6} \mapsto |x^2-6y^2|$. I think I proved this result with some common ...
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Field extensions and quotient fields

STATEMENT: Suppose that $F\subseteq E$ is a field extension of $F$. And assume $u\in E$ is transcendental over $E$. Then it readily follows that $F(u)\cong F(x)$, where $F(x)$ is the quotient field of ...
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1answer
27 views

Prove the automorphism given by $\phi \left(g\right)=\left(g^{-1}\right)^t$ is not an inner automorphism of $SL_n\left(R\right)$

Prove the automorphism given by $\phi \left(g\right)=\left(g^{-1}\right)^t$ is not an inner automorphism of $SL_n\left(R\right)$ Having no success with this question, I turn for your help =] I ...
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1answer
37 views

Proving a subgroup is normal

Problem Let $G$ be a group with $|G|=pm$, $p$ prime and $p \geq m$. Suppose there is $H$ subgroup of $G$ with $[G:H]=p$. Show that $H$ is normal. This problem was given to me in class just after ...
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2answers
21 views

Find the splitting field of a polynomial

The extension $\mathbb{Z}_p \leq \mathbb{F}_{p^n}$ is normal, as the splitting field of the polynomial $f(x)=x^{p^n}-x$ ($\mathbb{Z}_p$ is a perfect field therefore each polynomial is separable). So, ...
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1answer
29 views

Field of rational functions over $\mathbb{F}_p$

Let $K=\mathbb{Z}_p(x,y)$ be the field of rational functions of variables $x,y$ with coefficients in the field $\mathbb{Z}_p$, where $p$ is prime. Let $g(t)=t^p-x, h(t)=t^p-y \in K[t]$ and $E$ is the ...
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2answers
15 views

$\mathbb Z_{p^2}$ is not a non trivial semidirect product.

I am trying to prove that the group $\mathbb Z_{p^2}$ (p prime) is not a non trivial semidirect product. Since a group $G \cong K \rtimes H$ if and only if for all short exact sequences $$0 ...
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2answers
40 views

Showing that the product group of $G$ and $H$ satisfies the universal property for coproducts in the category of abelian groups $\mathbf{Ab}$

I'm working on another problem of Aluffi's Algebra. Given the category $\mathbf{Ab}$ of abelian groups, the problem is to show that for any two groups $G$ and $H$ the product group $G\times H$ ...
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2answers
50 views

For any group $G$, $|G/Z(G)| \neq 91$.

In Malik's Fundamentals of abstract algebra, one can find the following problem: Prove that for any group $G$, $\vert G/Z(G)\vert \neq 91$. This exercise is just ahead of Sylow's theorems. I've ...
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33 views

Show that the fields are equal

I have to show that $\mathbb{F}_{2^2}=\mathbb{Z}_2(a)$, where $a \in \mathbb{F}_{2^2}$ is of degree $2$ over $\mathbb{Z}_2$. $$$$ To show this do I have to take first an element of ...
3
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1answer
31 views

Direct sum of simple modules and Schur's Lemma

Suppose $M,N$ are two non-isomorphic simple $R$-modules. For $m,n\geq1$, is it true that $$ \text{Hom}_R(M^{\oplus m},N^{\oplus n})\cong\hat{0}\,? $$ I think it's true by Schur's Lemma. ...