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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Proof that group is commutative if every element is its inverse (feedback wanted)

This is one of my first proofs about groups. Please feed back and criticise in every way (including style & language). Axiom names (see Wikipedia) are italicised. $e$ denotes the identity element. ...
0
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0answers
12 views

subring of $\mathbb{Z}[i]$ and set $X$ of infinite cardinality that $\exists x \forall y \in X \,\,x^2|y^2$ but $\forall x \forall y \,\,x \not\mid y$

This is a question derived from A subring of the ring of Gaussian integers such that $a^2 | b^2$ does not lead to $a|b$ with infinitely many such cases. Is there a subring $R$ of Gaussian integers ...
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1answer
35 views

A question on short exact sequences.

The following is an excerpt from Atiyah-Macdonald on short exact sequences. I don't understand the part where the author says "Then $d(x'')$ is defined to be the image of $y'$ in Coker ($f'$)". Is ...
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3answers
28 views

Compound functions: one to one and onto

Let $f: A \to B$ and $g: B\to C$ be maps. If $g(f(x))$ is one-to-one and $f$ is onto, show that $g$ is one-to-one I'm really not sure how to prove this. Would someone be able to walk me through ...
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1answer
28 views

Determine if a function is a mapping

Given the function $f: \Bbb Q \to \Bbb Q$, explain why $f$ is a mapping: a) $$f\left( \frac pq\right) = \frac {3p}{3q} $$ b) $$f\left( \frac pq\right) = \frac {3p^2}{7q^2}- \frac pq $$ I ...
0
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1answer
38 views

A subring of the ring of Gaussian integers such that $a^2 \mid b^2$ does not lead to $a\mid b$ in infinitely many such cases

As the ring of Gaussian integers is a UFD, this means that $a^2 \mid b^2$ leads to $a\mid b$. Is there any subring of the ring of Gaussian integers with infinitely many elements such that ...
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0answers
18 views

regular representation of algebras

Let suppose we have universal enveloping algebra, what is the meaning of the notion of the right regular representation of that? How can we determine the right regular representation of universal ...
3
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2answers
60 views

In which Fields, does $x^n-x$ have a multiple zero?

In which Fields, does $x^n-x$ have a multiple zero? Attempt: Let $f(x) = x^n-x = x(x^{n-1}-1)$ and $f'(x) = nx^{n-1}-1$ If $f(x)$ has a multiple zero, then, $f(x)$ and $f'(x)$ have a common factor. ...
3
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2answers
17 views

Question about a passage in the Bicommutant Theorem's proof.

In the Averson's book, in the proof of the Von Neumann's Bicommutant theorem there is this passage: ($A $ is a self-adjoint algebra of operators in $L(H)$) "Let $\xi_1$ be an element of the Hilbert ...
2
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1answer
25 views

Group Theory Problem $T\subset S$

Let $S$ be a set, and $*$ an associative binary operation on $S$. Suppose there is an element $e\in S$ such that ($1$) $e*x=x$ and $x*e=x$ for all $x\in S$. (a) Prove that there is a unique element ...
2
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2answers
22 views

If $H$ is a normal subgroup of a finite group $G$ and $|H|=p^k$ for some prime $p$. show that $H$ is contained in every sylow $p$ subgroup of $G$

If $H$ is a normal subgroup of a finite group $G$ and $|H|=p^k$ for some prime $p$. show that $H$ is contained in every sylow $p$ subgroup of $G$ Attempt: $|H|=p^k \implies |G|=p^{n_1} q^{n_2} ...
3
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1answer
40 views

Groups - Prove that every element equals inverse of inverse of element

This is my first proof about groups. Please feed back and criticise in every way (including style & language). Axiom names (see Wikipedia) are italicised. We use $^{-1}$ to denote inverse ...
4
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1answer
31 views

Finding a cubic polynomial whose splitting field over $\mathbb{Q}$ equals $\mathbb{Q}(a)$ if $a$ is any of its roots

Question: Let $\alpha$, $\beta$ and $\gamma$ be the roots of a rational cubic polynomial $q$. Can we find a (non-trivial) example where the splitting field of $q$ over $\mathbb{Q}$ equals ...
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4answers
44 views

Show that $\mathbb{Q}(x)$ is not a homomorphic image of $\mathbb{Q}[y_1,\dots,y_n]$ for any $n$

How to show that the field of rational functions $\mathbb{Q}(x)$ is not a homomorphic image of $\mathbb{Q}[y_1,\dots,y_n]$ for any $n$? I thought proving this by contradiction. So suppose for some ...
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3answers
31 views

Find isomorphism between $\mathbb{Q}[T]/(T^2+3)$ and $\mathbb{Q}[T]/(T^2+T+1)$

The books states that the isomorphsim is $g(T)=2T+1$ and the identity when restricted to $\mathbb{Q}$. I would like some help to understand what the process is to find $g$.
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2answers
33 views

Show that if $G$ is a group of order $168$ that has a normal subgroup of order $4$ , then $G$ has a normal subgroup of order $28$

Show that if $G$ is a group of order $168$ that has a normal subgroup of order $4$ , then $G$ has a normal subgroup of order $28$. Attempt: $|G|=168=2^3.3.7$ Then number of sylow $7$ subgroups in $G ...
0
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1answer
25 views

Proof the existence of a normal subgroup

Let $K$ be a normal subgroup of $H/N$, and $N$ be a normal subgroup of $H$. Show that there is $M \lhd H$ such that $N \subset M$ and $K=M/N$. I have some difficulties to prove it.
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2answers
227 views

Linear algebra - Memorising proper definitions of homomorphism types

I am reading a book about linear algebra. On the basis of this book, I worked out the terminology below. Problem: To me, it looks like Wikipedia defines homomorphism differently. Apart from that: Do ...
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0answers
38 views

Ideal generated by a regular sequence

I need to prove that the ideal $$ I = (xz -y^2, x^2t^2 -yz^3, x^2yt^2 -xz^4) \subset R = \mathbb{K}[x,y,z,t]$$ is generated by a $R$-regular sequence. How can I do it? I don't know if this can ...
2
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3answers
71 views

If $G$ is a simple $f$ an homomorphism, and $A\lhd H$ is such that $[H:A]=2$, show $f(G) \subset A$

I'm stuck with the following problem. Can someone help me by providing a hint? Suppose $G$ is simple and let $f$ be an homomorphism between $G \to H$. If $\#G\ne2$, $A\lhd H$, and $[H:A]=2$. Then ...
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2answers
54 views

The multiplicative group of all the $2^n$-th roots of unity

Consider the multiplicative group $G$ of all the (complex) $2^n$-th roots of unity where $n=0,1,2...$. Which of the following statements are true? Every proper subgroup of $G$ is finite, $G$ has a ...
2
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3answers
32 views

Vector spaces - Non-uniqueness of element with property of scalar-multiplicative identity element?

I am dabbling in vector spaces, thinking about the axioms on Wikipedia. Notably, $$1 \mathbf{v} = \mathbf{v},$$ i.e. identity element of scalar multiplication (IEOSM), attracted my attention. I am ...
3
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0answers
29 views

If a finite group G is nilpotent then G/F is abelian, where F is the Frattini subgroup of G

Let $G$ be a finite nilpotent group. Consider $F$ the Frattini subgroup of $G$, that is, intersection of all maximal subgroup of $G$. Prove that $G/F$ is abelian. What I am trying that G/F is ...
0
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1answer
60 views

Field extension $\mathbb{Q} (\sqrt2)$; why is adding $\sqrt2$ not enough?

This an extract from Visual Group theory book section 10.5.1 page-234 ,which says that: What I can't understand from this extract is for what purpose do we need to add other elements except ...
4
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1answer
60 views

hint with an exercise algebra

I'm stuck with the following I hope someone could help me. Let $N$ a normal subgroup of $G$. Show that if $[G:N]=4$, exists a normal subgroup $M$ of $G$ s.t. $[G:M]=2$. My idea: Since $G/N$ has ...
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1answer
29 views

Field extensions: compute the degree of an extension.

I'm stuck with this problem. Let $F\subseteq E$ and $\gamma\in E$ is trascendental over $F$. Let $m$ be a positive integer. Show that $[F(\gamma):F(\gamma^{m})]=m$, where $[\quad:\quad]$ is the ...
4
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1answer
36 views

Does there exist an ideal in $\mathbb{Z}_4[x]$ which is prime but not maximal?

Question: Does there exist an ideal in $\mathbb{Z}_4[x]$ which is prime but not maximal? Thoughts: It seems to me that the ideal $(x)$ fails to be a prime ideal since $0 \in (x)= 2 * 2$ with $2 ...
3
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3answers
101 views

Prove that the division ring is commutative if for every $x$, $x^7=x$

I'm trying to solve a problem and I'm stuck. Here is the original problem: Let $A$ be a finite-dimensional algebra over a field $K$, such that for every $a\in A$, $a^7=a$. Show that $A$ is a ...
6
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2answers
133 views

Self studying higher mathematics?

I'm fairly well-versed in calculus but I would like to explore beyond calculus. I have looked into the basics of some topics in higher mathematics such as group theory and abstract algebra and they ...
4
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1answer
31 views

What is the relationship between the trace/norm of a quaternion and the definition in field theory?

I'm having some trouble figuring out the relationship between the trace/norm of a quaternion element and the definition of trace/norm in the extensions of vector spaces. According to my number theory ...
4
votes
3answers
95 views

What do the elements of the field $\mathbb{Z}_2[x]/(x^4+x+1)$ look like? What is its order?

Background: I'm looking at old exams in abstract algebra. The factor ring described was described in one question and I'd like to understand it better. Question: Let $F = \mathbb{Z}_2[x]/(x^4+x+1)$. ...
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1answer
36 views

Smallest possible odd integer that can be the order of a non-Abelian group.

Smallest possible odd integer that can be the order of a non-Abelian group. Attempt: A non abelian group means $Z(G) \subset G$ . Hence, it suffices to find the smallest odd integer $n$ such that ...
2
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1answer
46 views

The smallest composite integer $n$ such that there is a unique group of order $n$?

We need to find the smallest composite integer $n$ such that there is a unique group of order $n$. Attempt: Let us suppose that $n= ab$ is a composite integer where $a,b$ are integers such that ...
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0answers
35 views

Irreducible polynomial

Does there exist an irreducible polynomial over a field K with two roots $a,b$ and $k\in K$ such that $a=b+k$ ? This can't happen if K is of characteristic $0$ , but can it happen if K is of ...
0
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1answer
20 views

Semiring that has unique factorization except zero

In a ring, there is unique factorization domain. Then is there a similar concept in semiring - that is a commutative semiring that has unique factorization for every element except zero? If so, what ...
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3answers
57 views

Prove that $I$ is a maximal ideal of $\mathcal A$. [duplicate]

Please, give-me a hint to prove this proposition: Let $\mathcal A$ be the ring of all continuous real functions (with the usual operations of sum and multiplication) defined on the interval ...
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2answers
62 views

Are $3 \Bbb Z/6 \Bbb Z$ and $\Bbb Z_3$ isomorphic?

I'm trying to prove whether or not these to groups are isomorphic.
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0answers
27 views

An integral domain that has square of prime elements share same greatest common factor, whil [on hold]

Is there any numerical integral domain, not involving monomials or polynomials that has square of prime elements share same greatest common factor $g$, while product $P$ of two different prime ...
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1answer
30 views

Minimal polynomial and field extension

If the degree of a field extension $[\mathbb{Q}(\alpha):\mathbb{Q}]=n\gt 1$ and $\alpha$ is a root of a monic polynomial $f \in \mathbb{Q}[T]$ and the degree of $f$ is $n$. Does the above imply ...
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Annihilator of annihilator of annihilator of a submodule

Let $R$ be a commutative ring and M be an $R$-module. The question is whether for every submodule $N$ of $M$ the equality ...
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0answers
15 views

Degree of the separable closure of the residue field of a complete field

I'm trying to prove a corollary on ramified extensions but I don't know if my thoughts are true. Let $L$ be a finite extension of a complete discrete valuation field $F$, let ...
2
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1answer
57 views

How to split up $X^{20}-1$ in $\mathbb{F}_3[X]$

I try to split $X^{20}-1$ into irreducible polynomials in $\mathbb{F}_3[X]$. The first thing I saw is that $1$ is a root. Second, $-1$ must be one too. I have taken the derivative $20X^{19}$ to ...
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1answer
27 views

Number of distinct subgroup of given group [on hold]

Let $ G $ be the group of all automorphisms of field $F_{3^{100}}$ that consist of $3^{100}$ elements. Then number of distinct subgroup of G equal to 1. 4, 2. 3, 3. 100, 4. 9.
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2answers
40 views

In $S_4$, what does the expression for the cyclic group $\langle (13),(1234)\rangle$ mean?

In $S_4$, what does the expression for the cyclic group $\langle (13),(1234)\rangle$ mean? I apologize if this is too basic, but I haven't come across such an expression anywhere in my book. Also, ...
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2answers
33 views

Question about the symmetric group

How do I prove that if $f\in S_k$ and $f^n=f^m=Id$, then $f^d=Id$ where $d=gcd(n,m)$? I tried writing $f^{dn_1}=f^{dm_1}=f^0$ but this does not lead anywhere. I think I should use that $n_1$ and ...
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25 views

Soluble group algebras and centralizers

Let $K$ be field with $\mathrm{char}(K) = p > 0$ and $G$ a finite group such that $KG$ is soluble. Then the $p$-Sylow-subgroup of $G$ is normal and contains the derived group of $G$ and let $H$ be ...
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Examples of reduced associative algebras

An associative $K$-algebra A is called reduced if $A/rad(A)$ has no nilpotent elements. It can be shown that this is equivalent to that $A/rad(A)$ is a isomorphic to a direct sum of division algebras. ...
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1answer
39 views

What does “generate freely” mean?

Given a number field $K$ (i.e. $\mathbb Q\le\ K\le\mathbb C$, $[K:\mathbb Q]=n$), the relative number ring is $R=\mathbb A\cap K$, where $\mathbb A$ is the ring of the algebraic integers in $\mathbb ...
2
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1answer
49 views

Valuation rings of $k(X)$

My question is how to determine all valuation rings of the field $k(X)$ containg the field $k$. I want to show that if $V$ is a valuation ring of the field $k(X)$ and $\neq k(X)$ then ...
0
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2answers
39 views

$\dim (U_1\cap U_2)\ge \dim U_1+\dim U_2-\dim V$

I'm reading the excellent and incredible well-written book: Algebraic Function Fields and Codes by Henning Stichtenoth. I don't remember this theorem in my linear algebra course, maybe this is a ...