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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Proof of a result on closed subgroups of Galois group

Let $M\supseteq K$ be an algebraic normal extension. Then its Galois group is profinite. I have been told that in this hypothesis, if $H\leq G$, then $H''=H\iff \bar H=H$, where ...
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Conjugates of an $r$-cycle in $S_n$

How many conjugates does a cycle of length $r$ have in the permutation group $S_n$? I tried to find them but failed.
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Subgroup of group is normal

This question came in the exam today, unfortunately I couldn't answer it. The question said: Proof whether or not this is a true statement, stating the reason. Subgroup of group is normal I ...
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1answer
29 views

In $S_{3}$ what is the group generated by $(123)$?

In $S_{3}$ what is the group generated by $(123)$? Is there a way to find the elements of the group generated by $(123)$?
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1answer
25 views

Show that an $R$-module homomorphism $\alpha:A \to B$ is injective.

I am working on an exercise on injective modules: Show that an $R$-module homomorphism $\alpha:A \to B$ is injective if the induced map Hom$_R(B,Q)\to$ Hom$_R(A,Q)$ is surjective for all injective ...
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Question about direct product of two groups.

Let $G=\mathbb{Z}_n \times \mathbb{Z}_m$ and $d=p^k$ for some prime $p$ such that $d$ divides both $n$ and $m$. Then $G$ has exactly $d\phi(d)+[d-\phi(d)]\phi(d)$. For example consider the group ...
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0answers
24 views

Understanding a quotient ring of continuous functions

In trying to understand another questions answer(to a question I asked), I realized that my fundamental lack of knowledge was in regards to the following question: In terms of functions, what does ...
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9 views

Let $G$ be a group ,and $d \mid ord(G)$ ,let $ED:=\{g\in G| g^d=e\}$ show that $d \mid ord(ED)$ [duplicate]

Let $G$ be a group ,let $d$ be a number such that $d \mid ord(G)$ ,let $ED:=\{g\in G| g^d=e\}$ show that $d \mid ord(ED)$ Any ideas?
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1answer
61 views

How to learn commutative algebra?

I want to learn commutative algebra from scratch. I was wondering, as you guys are experts in mathematics, what you think is the best way to learn commutative algebra? Is there any video course ...
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2answers
39 views

Maximal ideals of the ring of all continuous functions

Let $R$ be the ring of all continuous functions from the interval $[0,1]$ to $\Bbb R$. For each $a\in[0,1]$ let $$A_a=\{f\in R \mid f(a)=0\}$$ Now firstly, this is part of an assignment problem, ...
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21 views

The relationship of subnormal subgroups and modular subgroups of a finite group.

Let $G$ be a finite group, a subgroup $H$ of $G$ is called subnormal if it's a term of a composition series of $G$, and is called modular if it's a modular element of the subgroup lattice $L(G)$. My ...
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25 views

Prove that a transformation of the identity functor of a Group $G$ (seen as a category) into itself is just an element of the center of $G$

I want to prove the follow: Suppose $G$ is a group seen as a category, prove that a transformation of the identity functor of $G$ into itself is just an element of the center of $G$. I'm not ...
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1answer
28 views

Groups occuring as derived subgroups.

I want to prove this problem but I have no idea how to start it. If you know please hint me, thanks. Suppose that $G$ is a group that has subgroup which is cyclic, characteristic and not in the ...
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Factorisation of Characteristically Simple Group

Please am working on a project "Factorization of characteristically simple group". And it has been really difficult to locate articles relating to that, even on the web. So i want to know if any one ...
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1answer
19 views

Find an ideal of $R = \mathbf{Z}/12\mathbf{Z} = \{0,1,2,\dots,11\}$ with two elements

Let $R = \mathbf{Z}/12\mathbf{Z} = \{0,1,2,\dots,11\}$. Find an ideal $I$ of $R$ which consists of two elements. How many elements does $R/I$ have? I thought the ideals would be $\{0\}$, ...
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1answer
34 views

Show that the subset $\overline{I} = \{\overline{x}:x \in I\}$ is an ideal.

Assume that $I$ is an ideal of the ring $\mathcal{O}_d = \left\{ \begin{array}{ll} \mathbb{Z} [\sqrt{d}] & \text{ if } d \text{ is even } \\ \mathbb{Z} [ \frac{1 + ...
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3answers
25 views

Show that if $A, B$ are groups, then $A \times B$ is solvable if and only if both $A$ and $B$ are solvable? [on hold]

Show that if $A, B$ are groups, then $A \times B$ is solvable if and only if both $A$ and $B$ are solvable? Why is obvious that if $A$ and $B$ are solvable then $A \times B$ is solvable?
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59 views

Proof that if $a,b \in G$ and $a^4b = ba$ and $a^3 = e$ then $ab = ba$

I tried to prove one of the examples in my Abstract Algebra book that stated: Prove that if $a,b \in G$ and $a^4b = ba$ and $a^3 = e$ then $ab = ba$ I went about just saying that $a^4b = ba ...
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2answers
30 views

Prove that if $Z(G) = \lbrace x \in G: gx = xg \text{ for all }g\in G\rbrace$ then $Z(G)$ is a group

So my challenge is: Prove that if $Z(G) = \lbrace x \in G: gx = xg \text{ for all }g\in G\rbrace$ where $G$ is a group, then $Z(G)$ is a group For the identity, $e$ clearly is in $Z(G)$ and in ...
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32 views

Show that $f: \mathbb{Z} \to n\mathbb{Z}$ defined as $f(m) = mn$ is bijective

Let $n \in \mathbb{N}$. Show that $f: \mathbb{Z} \to n\mathbb{Z}$ defined as $f(m) = mn$ is bijective such that $f(m_1 + m_2) = f(m_1) + f(m_2)$, $\forall m_1, m_2 \in \mathbb{Z}$ To be bijective, ...
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100 views

Show that $x^{3}-3$ irreducible over $\mathbb{Q}(\sqrt{-3})$

Is there a slick way to show that $x^{3}-3$ is irreducible over $F= \mathbb{Q}(\sqrt{-3})$? What I did seems kind of convoluted (showing directly that there is no root in F). Thanks
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1answer
34 views

algebraic integers of $\mathbb{Q}(\sqrt{d})$

Assume that $d$ is square-free. What is the set of algebraic integers in $\mathbb{Q} \left(\sqrt{d} \right) = \{a + b \sqrt{d}:a,b \in \mathbb{Q} \}$? The algebraic integers in ...
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1answer
31 views
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32 views

$\gcd(a,n)\neq 1 \implies $ there is $b$ such that $ab\equiv 0 \pmod{n}$

I have that $\gcd(a,n)\neq 1$ ($a$ and $n$ are not coprime). Then, somehow, I need to prove that exists an $b$ such that $$ab\equiv 0 \pmod{n}$$ What I did: $$ab\equiv 0 \pmod{n}$$ is the same ...
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1answer
26 views

Why does it mean that $n$-th variable is removable?

I'm reading the proof for "the fundamental theorem of symmetric polynomials" and I have a trouble with it (http://en.m.wikipedia.org/wiki/Elementary_symmetric_polynomial) Let $P(X_1,...,X_n)$ be a ...
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1answer
56 views

Extension $\mathbb{Q}(\sqrt{-5}, i)/\mathbb{Q}(\sqrt{-5})$, splitting.

In the extension $\mathbb{Q}(\sqrt{-5}, i)/\mathbb{Q}(\sqrt{-5})$, must principal prime ideals of $\mathbb{Z}[\sqrt{-5}]$ necessarily split into 2? Must nonprincipal prime ideals not split? ...
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26 views

How to get real irreducible matrix representations from the complex irreducible matrix representations?

I'm trying to get real symmetry adapted orbitals for molecules with icosahedric symmetry (point groups $I$ and $I_h$) using the complete projector operator (truly projector if i=j): \begin{equation} ...
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2answers
45 views

Algebra and Maximal ideal.

I am trying to solve the following problem. If $ \mathcal{K}$ is a field and $a_1,a_2,\dots,a_n \in \mathcal{K}$. Prove that $(x_1-a_1,x_2-a_2,\dots,x_n-a_n)$ is a maximal ideal in ...
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69 views

Non-abelian Group with infinite exponent in which every proper subgroup has finite exponent

can you find a Non-abelian Group with infinite exponent in which every proper subgroup has finite exponent?
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30 views

Fundamental theorem of Algebra using ideas of complex singularities

Below is an excerpt from Arnold's Theory of Catastrophes (I haven't got an American edition, so translating from Russian). Where I can read about it in more detail? Not only regarding polynomials. ...
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25 views

Looking for a non trivial homomorphism II [on hold]

Is there a non trivial homomorphism $f: SU(2) \to Diff(S^1)$?
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1answer
60 views

Looking for a non trivial homomorphism I

Is there a non trivial homomorphism $f: SU(2) \to O(2)$? Is there a concrete description of $Hom(SU(2), O(2))$?
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15 views

Is $\alpha$ a norm in the extension $K(\sqrt[n]{\alpha})$?

I'm having trouble wrapping my head around this. $K$ is a field of characteristic zero containing all $n$th roots of unity, and $\alpha \in K$. Let $L = K(\sqrt[n]{\alpha})$, $\mu$ the minimal ...
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Why is an $n$th power a norm in a Kummer extension?

Let $F$ be a $p$-adic field containing the $n$th roots of unity. Then by Kummer theory, $[F^{\ast} : F^{\ast n}]$ (which is finite) is equal to the cardinality of $\textrm{Gal}(E/F)$, where $E$ is ...
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1answer
14 views

For any monomial ordering, $1\leq m$ for any monomial $m$

Let $R$ be a ring. Let $\leq$ be a well-ordering on the set of (monic) monomials in $R[X_1,...,X_n]$. Then, $\leq$ is said to be a monomial ordering iff $mm_1\leq mm_2$ whenever $m_1\leq ...
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2answers
43 views

A normal subgroup so that any homomorphism into a $p$-group is trivial on it. [on hold]

Problem Let G be a finite group of order $n$ and $p|n$. Show that there is a unique normal subgroup $N$ satisfying the following property: (1)$G/N$ is a $p$-group (I guess it can be trivial group). ...
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3answers
53 views

A finite group which has a unique subgroup of order $d$ for each $d\mid n$.

Problem Suppose G is a finite group of order $n$ which has a unique subgroup of order $d$ for each $d\mid n$. Prove that $G$ must be a cyclic group. My idea: I try to prove it by induction. Let ...
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1answer
53 views

How to prove that the ring of algebraic integers is a Bézout domain?

I was told that the ring of all algebraic integers (that is, the complex numbers which are roots of a monic polynomials with integral coefficients) is a Bézout domain. But I have no idea how to prove ...
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1answer
21 views

Show that a representation of a finite group is isomorphic to its dual if its character takes only real values

This appeared as a part of showing that a representation of a finite group is isomorphic to its dual if and only if its character takes only real values. The "only if" part was easy to show. For the ...
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2answers
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Algebra generated by a single element over an infinite field, does $A_1 \times \cdots \times A_r$ has the same property?

Let $K$ be an infinite field and $A_1, \ldots, A_r$ algebras over $K$ finite dimensional and such $\forall i = 1, \ldots, r \ \exists x \in A_i : A_i = K[x]$ (I think we say that $A$ is finitely ...
3
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1answer
68 views

Matrix and field extension

It is given that $F\subset K$ are fields. $A$ is a matrix of size $n\times n$ over $K$. I need to prove that there exist $c_1,\ldots,c_k\in K$, linearly independent over $F$, and matrices ...
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1answer
35 views

Proof that if $H_1 \leq G$ and $H_2 \leq G$ then $H_1 \cap H_2 \leq G$

I am trying to prove that Prove that if $H_1 \leq G$ and $H_2 \leq G$ then $H_1 \cap H_2 \leq G$ Unlike this question: Prove $H_1 \cap H_2 \le H_1 $ when $H_1, H_2 \le G$ and $H_1$, $H_2$ are ...
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1answer
50 views

What is orthogonal group $O(1)$?

I know that $O(2)$ is the group of 2x2 orthogonal matrices, but how can we extend the meaning of group and orthogonal to $O(1)$?
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40 views

Show that if for $a \in \mathbb{Q}$ with $0 = f(a) $ for a monic polynomial $f(x)\in \mathbb{Z}[x] $, then $a \in \mathbb{Z}$

I. Show that if for $a \in \mathbb{Q}$ with $0 = f(a) \in \mathbb{Z}[x] $ for a monic polynomial, then in fact $a \in \mathbb{Z}.$ II. The set of algebraic integers of $\mathbb{Q}$ is ...
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2answers
66 views

Is the left translation $T_a(x) =ax $ a homomorphism?

I apologize if this is a super basic question but I was reading Lang's undergraduate algebra book and it says that the following function is a homomorphism: $$T_a(x) = ax$$ The way I would check if ...
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1answer
42 views

Looking for a direct proof that all maximal ideals of $\mathbb C[x_1,x_2,…,x_n]$ are generated by $n$ linear polynomials

Without using Hilbert's Nullstelensatz , can we directly prove that all maximal ideals of $\mathbb C[x_1,x_2,...,x_n]$ is of the form $\langle x-a_1,x-a_2,...,x-a_n \rangle$ ? It is easy to prove it ...
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4answers
53 views

How do we find the minimal polynomial of $\alpha = a + b \sqrt{d}$ over $\mathbb{Q}$?

Let $\alpha = a + b \sqrt{d} \in \mathbb{Q} \left(\sqrt{d} \right) = \{a+b \sqrt{d}:a,b \in \mathbb{Q} \}.$ The minimal polynomial $m(x)$ of an algebraic number $\alpha \in \mathbb{C}$ is the monic ...
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1answer
42 views

Is there a function over $\mathbb{Z}_p$ that is never linear?

Let $p$ be a prime. I wonder if there is a function $f$ that satisfies the following rule: Whenever $$z_1 + \dots + z_c \equiv cx \mod p$$ (where $1 < c < p$, and $z_j, x \in \mathbb{Z}_p$) ...
2
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1answer
45 views

Any curve of genus three is either hyperelliptic or trigonal?

A curve $C$ is said to be trigonal if it admits a rational function $f: C \to \mathbb{CP}^1$ of degree $3$. Is any nonsingular plane curve of degree four trigonal? Can the map be chosen so that it has ...
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1answer
72 views

Prove the isomorphism of categories $Fun(\mathcal{A}\times\mathcal{B},\mathcal{C})\cong Fun(\mathcal{A},Fun(\mathcal{B},\mathcal{C})),$

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 ...