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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PID vs UFD $\Bbb{Z}\sqrt{-5}$?

Is $\Bbb{Z}[\sqrt{-5}]$ an example of a ring which is a PID but not UFD?
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24 views

Every nonabelian group of order divisible by 6 contains a subgroup of order 6

I have a question I was hoping for help on: Prove or disprove every nonabelian group of order divisible by 6 contains a subgroup of order 6 I would guess that this statement is true based on a ...
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1answer
18 views

Prove that $\forall a,b\in\mathbb{C} \exists c,d \in \mathbb{C}: c+d = a \land cd = b$

Prove that: $\forall a,b\in\mathbb{C} \exists c,d \in \mathbb{C}: c+d = a \land cd = b$ We just learned about the characteristic/minimal polynomial and diagonalization but I am not sure if it has ...
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1answer
27 views

Rotman sylow excercise question. Did he ommit finite?

Let $P$ be a sylow $p$-subgroup of $G$. Let $N(P) \leq H $ .Prove $H=N(H)$.Did Rotman ommit that $G$ is finite? This is excercise 4.11 in the fourth edition.
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19 views

The basis for orthogonal complement of a subspace [on hold]

The following is my problem, thank you so much.
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20 views

When do (universal-algebraic) varieties have (enough) injectives?

Following Zhen Lin's suggestion in my previous question, I ask this question separately. Suppose $\mathcal{V}$ is a variety in the sense of universal algebra. considered as a category. When does ...
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2answers
49 views

Are varieties cocomplete?

Consider a variety $\mathcal{V}$ in a sense of universal algebra, i.e. algebras of some fixed signatures described by a set of identities. Then $\mathcal{V}$ can be thought of as a category with ...
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37 views

If $\overline{f}(x)$ is irreducible in $\mathbb{Z}_p[x]$ then $f(x)$ is irreducible in $\mathbb{Z}[x]$

Let $f(x)=a_0+\dots +a_n x^n \in \mathbb{Z}[x]$. Let $p$ be a prime with $p \nmid a_n$. We define $\overline{f}(x)=\overline{a_0}+\dots +\overline{a_n} x^n \in \mathbb{Z}_p[x]$ How can I show that ...
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1answer
11 views

What is the relation between $Irr(a, F)$ and $Irr(a, K)$?

We have that $F \leq K \leq L$ and $a \in L$. If $a$ is algebraic over $F$ then it is also algebraic over $K$. What is the relation between $Irr(a, F)$ and $Irr(a, K)$? Let $Irr(a, K)=p(x) \in K[x]$ ...
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1answer
19 views

Direct product of simple non-abelian groups

Let $G$ be a group, and let $K$ be a normal subgroup of $G$ which is a direct product of simple non-abelian groups. I wanted to prove that $K=C_K(H)[K,\, H]$ for every subgroup $H$ of $G$. Is this ...
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58 views

$\sqrt{5} \in \mathbb{R}$ is algebraic over $\mathbb{Q}$

I have to show that $\sqrt{5} \in \mathbb{R}$ is algebraic over $\mathbb{Q}$ and then I have to find $Irr(\sqrt{5}, \mathbb{Q})$. How can I show that $\sqrt{5} \in \mathbb{R}$ is algebraic over ...
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20 views

Why does it stand that #$\mathbb{Z}_p(a)=p^n$?

If $\mathbb{Z}_p \leq K$ an algebraic extension, then $K$ has the identity $$\forall a \in K, \exists b \in K \text{ with } a=b^p$$ The proof is the following: Let $a \in K$. We take $\mathbb{Z}_p ...
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0answers
25 views

How to show that $E=\mathbb Q(\alpha)$

Let $p\neq 2$ a prime number, $\zeta=e^{\frac{2i\pi}{p}}$ and $\alpha=2\cos\left(\frac{2\pi}{p}\right)$. We consider the field extension $F=\mathbb Q(\zeta)$ and $E=F\cap \mathbb R$ of $\mathbb Q$. I ...
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1answer
29 views

Example of non-commutative ring with exactly 2014 two sided-proper ideals.

find a non-commutative ring with exactly 2014 two sided-proper ideals.find a ring with exactly 2014 pairwise non-isomorphic irreducible modules. If it was the commutative ring i would have thought of ...
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29 views

If a certain ideal is radical or not

Let $n \in \mathbb{N}$ and let $I_{n}$ be an ideal in the polynomial ring $\mathbb{C}[x_{1},...,x_{n}]$ with the following properties: I is generated by a (finite) number of polynomials which are ...
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1answer
17 views

Minimal right ideals

Let $I$ be a minimal right ideal of a ring $R$ with $1$. If $r\in R$, could we say that $rI$ is zero or a minimal right ideal? I assumed a right ideal $J$ in $rI$ and intersecting it with $I$ got a ...
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1answer
19 views

How do we find the embeddings?

In my notes there is the following example: $$\mathbb{Q}(\sqrt{2}) \overset{\widetilde{\sigma}}{\longrightarrow}\mathbb{R}\\ | \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ | \\ \mathbb{Q} \overset{\sigma=id ...
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2answers
46 views

Suppose G is a group, p is prime , Then the number of elements of G of order p is multiple of (p-1) [on hold]

I need Help . "Suppose $G$ is a group, $p$ is prime , Then the number of elements of G of order $p$ is multiple of $(p-1) $". Give me any advise or note
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27 views

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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1answer
26 views

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 elements $\mathbb{Z}_2$. Also, how would ...
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28 views

infinite version of lagrange theorem [duplicate]

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
40 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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87 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
30 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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22 views

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
35 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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48 views

If $n\mid m$ prove that the canonical surjection $\pi: \mathbb Z_m \rightarrow \mathbb Z_n$ is also surjective on units

Not sure if this is the right proof (i found it online): Since $n\mid m$, if we factor $m = p_1^{\alpha_1}p_2^{\alpha_2}\cdots p_k^{\alpha_k}$, then $n = p_1^{\beta_1}p_2^{\beta_2}\cdots ...
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2answers
30 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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27 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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25 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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1answer
28 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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130 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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2answers
48 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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36 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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2answers
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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38 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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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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30 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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21 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 ...
2
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1answer
50 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 ...
3
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1answer
61 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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62 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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16 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 ...
5
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1answer
126 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 ...
2
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1answer
39 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 ...