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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Is $\mathbb R \times \mathbb Q$ a principal ideal domain

I have a similar question to this one: $\mathbb Z\times\mathbb Z$ is principal but is not a PID Is $\mathbb R \times \mathbb Q$ a principal ideal domain/ring (that is - is every ideal in $\mathbb R ...
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0answers
4 views

Sufficient and necessary conditions for representation of a concatenation structure.

Given a structure $\mathcal{A} = (A, \succsim, \sqcup)$, where $A$ is a non-empty set, $\succsim$ is a weak order, $\sqcup$ is a binary operation on $A$, let $\mu$ be an order-preserving mapping from ...
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0answers
9 views

Normal semilattices of groups!

Can someone help me to get an answer to this question: Let $S$ be a Clifford semigroup and $S'$ sub-semigroup of $S$ and if $S'_{e_{n}}$ is normal in $S_{e_{n}}$, what can we say about the $S'$( is ...
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2answers
47 views

Prove there is no simple group of order $729$

Let G be a group of order $729$. $729 = 3^6$ so by Sylow's Theorem G has a Sylow $3$-subgroup of order $729$. And there are $x$ of them. $r \equiv 1 \pmod 3$ and $r\ |\ 1$. So $x=1$? Is this ...
1
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1answer
23 views

Let $R$ be a ring. Let $I\lhd R$ and fix $n\in I$ if $n$ is the unit of $R$. Show $R=I$

Let $R$ be a ring. Let $I\lhd R$ (that is $I$ is an ideal of the ring) and fix $n\in I$ if $n$ is the unit of $R$. Show $R=I$. Here is my attempt at an answer: We aim to show $I \subseteq R$ and $R ...
1
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1answer
26 views

The root system of $sl(3,\mathbb C)$

I want to determine the root-system of the lie algebra $sl(3,\mathbb C)$. Does someone know a good (and complete) reference for this problem? I know that the root-system is $A_2$ but I want to see a ...
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1answer
20 views

Help with constructing a certain ring in GAP III

I need to construct the following ring in GAP: $$Z_4(u) / \langle u^2-2u=0 \rangle $$. This is what I tried and it didn't work: ...
0
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1answer
24 views

Let $d \in \mathbb{Z}$, $d > 1$. Determine all the ideals of $\mathbb{Z}/d\mathbb{Z}$ which are prime or maximal

Let $d \in \mathbb{Z}$, $d > 1$. Determine all the ideals of $\mathbb{Z}/d\mathbb{Z}$ which are prime or maximal I know that $m\mathbb{Z}/md\mathbb{Z} \cong \mathbb{Z}/d\mathbb{Z}$ as rings ...
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1answer
15 views

Help with constructing a certain ring in GAP II

I need to construct the following ring in GAP: $$F_2(u_1,u_2) / \langle u_1^2=u_2^2=0,u_1u_2=u_2u_1 \rangle $$. This is what I tried and it didn't work: ...
4
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1answer
25 views

Is it true if $R = mZ/mdZ$ is isomorphic to $Z/dZ$, then it must have a unit element?

Is it true if $R = mZ/mdZ$ is isomorphic to $Z/dZ$, then it must have a unit element? This is a question I ask myself, but I'm not certain of this answer. Is anyone could explain to why this is (or ...
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1answer
25 views

What are the ideals of the ring $\mathbb{Z}[x]/(2,x^3+1)$?

What are the ideals of the ring $\mathbb{Z}[x]/(2,x^3+1)$? I'm stuck at how to determine what ring this ring is isomorphic to?
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1answer
40 views

Computing $[\mathbb Q(\sqrt[4] 2):\mathbb Q(\sqrt2)]$

What is $[\mathbb Q(\sqrt[4] 2):\mathbb Q(\sqrt2)]$? I think it's 2 but I'm not sure why.
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0answers
17 views

Prime and Maximal Ideals of $\mathbb{Z}[x]$ [duplicate]

Consider $R=\mathbb{Z}[x]$. Also let $p$ be a prime. Then we want to find all the prime and maximal ideals of $\mathbb{Z}[x]$. The prime ideals are $(0), (p), (x)$ and $(ap + bx)$. Then we see that ...
1
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0answers
34 views

$ax=0$ if and only if $a=0$ or $x=0$ [duplicate]

Prove that $ax=0$ $\Leftrightarrow$ $a=0$ $\lor$ $x=0$, where $a$ is a scalar from a field and $x$ is an element of the vector space on this field. I would like a hint or maybe a solution to prove ...
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0answers
13 views

Prove an isomorphism between this two algebraic objects [duplicate]

I want to prove that $\bar{K}[V]/M_p \simeq \bar{K}$ where $K$ is a field, $\bar{K}$ is its algebraic closure and $$\bar{K}[V]=\bar{K}[x_1,...,x_n]/I_V,$$ where $I_V$ is the ideal attached to a ...
1
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1answer
19 views

Commutator subgroup $[H,K]$, $H, K$ subgroups of a group

How could I show that $[H,K]$ is a normal subgroup of $\langle H, K \rangle$? Also that if $H$ is generated by $X$ and $K$ is generated by $Y$, then $[H,K]=\langle g[x,y]g^{-1} | x \in X, y\in Y, g\in ...
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0answers
11 views

For the special class of $T$-subgroups in a certain quotient group the normal subgroups intersect certain subgroup

Let $G$ be a finite group and $U \le G$ be a subgroup of odd order which has index two in its normalizer and $U^g \ne U$ implies $U^g \cap U = 1$. Write $N_G(U) = TU$ with $T = \langle t \rangle$ for ...
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1answer
14 views

Splitting fields and intermediates [on hold]

Prove that if $F$ is a splitting field of $S$ over $K$ and $E$ is an intermediate field, then $F$ is a splitting field of $S$ over $E$.
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1answer
19 views

$F$ need not be normal over $K$ [on hold]

Prove that if $F$ is normal over an intermediate field $E$ and $E$ is normal over $K$, then $F$ need not be normal over $K$. No clue.
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0answers
35 views

Collective name for algebraic structures

I am doing a thesis about various algebraic structures, primarely about groups, rings and modules (with maybe hint of algebras). However always having type out ALL of them constantly gets very tedious ...
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1answer
35 views

P.I.D. and a nontrivial ideal, Quotient ring has finitely many ideals [on hold]

A ring $R$ is a P.I.D. Let $I$ be a nontrivial ideal in $R$. Prove that $R/I$ has finitely many ideals.
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0answers
30 views

Classify all finite groups with property [on hold]

Classify all finite groups $G$ with the following property: for every $H\vartriangleleft G$ there exists $K<G$ such that $G/H$ is isomorphic to $K$. My poor abstract-algebraic imagination doesn't ...
2
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0answers
35 views

defining gcd on rings

I see that in most textbooks they say let $R$ be an integral domain and start defining the greatest common divisor. My question is, can gcd's be defined on just commutative rings without an identity?
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0answers
9 views

How are the embeddings of a subfield of a Galois extension $K$ related to the embeddings of $K$?

Suppose we have a Galois extension $K/\mathbb{Q}$, then all embeddings of $K$ into $\mathbb{C}$ (or $\mathbb{R}$) are determined by the Galois group $G=\text{Gal}(K/\mathbb{Q})$. That is if we let ...
4
votes
3answers
41 views

Inverse of $f: \mathbb{Z}_{21} \times \mathbb{Z}_{10} \to \mathbb{Z}_{210}$ such that $f(a,b)= 10a +21b$

Let $f: \mathbb{Z}_{21} \times \mathbb{Z}_{10} \to \mathbb{Z}_{210}$ such that $$f(a,b)= 10a +21b.$$ We have that $f$ is an isomorphism, but how does one go about finding explicitly the inverse ...
0
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0answers
14 views

Subgroup of order $9$ and $4$ in $\langle \alpha, \beta \rangle$

Let $$\alpha = (1,2,3,4,5)(6,7,8)(9,10,11)$$ $$\beta = (1,2,3)(4,5)(6,9,7,10,8,11)$$ We have that $\langle \alpha \rangle \cap \langle \beta \rangle = \{id\}$. So $$ord \langle \alpha, \beta \rangle ...
2
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1answer
14 views

Example of inverse semigroup with at least two idempotent elements

We say that the semigroup $S$ is inverse semigroup if for any $s\in S$ there is a unique $t\in S$ such that $sts=s,\ tst=t$. Suppose that $E(S)=\{e:\ e\in S,\ e^2=e\}$ and define $$s\sim ...
1
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1answer
44 views

A subset of a polynomial ring and its ideal. [duplicate]

Let $P=K[x_1,\dots,x_n]$ be a polynomial ring over a field $K$ and $I = (f)$ be a principal ideal in $P$ generated by $f \in P - \{0 \}$. Moreover let $L \subset \{x_1, \dots, x_n \}$ and $\hat{P} ...
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0answers
30 views

Help me with this Group Question

Let Sn := Bij({1, . . . , n}) the symmetric group with n elements equipped with composition of functions. (i) Show that Sn is a group. (ii) Let (e1, . . . , en) be the canonical base of Rn and Mn(R) ...
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1answer
44 views

Find the greatest common divisor of pairs of polynomials

I'm trying to find the greatest common divisor of $$p(x)=7x^3+6x^2-8x+4$$ and $$q(x)=x^3+x-2$$ where both $p(x),q(x)\in\mathbb{Q}[x].$ And if $d(x)=gcd(p(x),q(x)),$ I need to find two polynomials ...
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votes
1answer
29 views

When does normal maximal subgroup have prime index?

Given finite group $G$, a normal maximal subgroup $H$, when is $[G:H]$ a prime? If $G$ is nilpotent, then the statement is true. But I am not sure about other $G$. Is there any counter-example ...
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0answers
25 views

$K$-affine $Hom$ functor relating to Polynomial Maps (Exercise in _Algebra: Chapter 0_ by Aluffi)

I'm currently learning some category theory and algebraic geometry, the basics at least. In Algebra: Chapter 0 by Aluffi, there is an exercise describing the relationship between morphisms of affine ...
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2answers
22 views

$f\in L[x,y]$ such that $f(x,0)=0$ implies $f=y g$ with $g\in L[x,y]$?

Suppose $L$ is an infinite field (or even algebraically closed; I'm not sure if it is necessary to add that hypothesis). If we have a polynomial $f(x,y)\in L[x,y]$ and $f(x,0)\equiv 0$, does that ...
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votes
1answer
25 views

Proving group and Morphisms of groups [on hold]

Let Sn := Bij({1, . . . , n}) the symmetric group with n elements equipped with composition of functions. (i) Show that Sn is a group. (ii) Let (e1, . . . , en) be the canonical base of Rn and Mn(R) ...
3
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2answers
35 views

If $|F| = 8$ and $K$ is a subfield, show that $K = F$ or $K = \{0, 1\}$.

Let $K$ be a subring of a field $F$. If $|F| = 8$ and $K$ is a subfield, show that $K = F$ or $K = \{0, 1\}$. [Hint: Lagrange]. Lagrange's Theorem: If $H$ is a subgroup of a finite group $G$ I Then ...
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2answers
36 views

The center of a group

prove that for any group G, Z(G)=$\bigcap_{x\in G} C_{G}(\{x\})$ . In addition, show that if H$\subset$G , then $C_{G}(H)=\bigcap_{x\in H} C_{G}(\{x\})$ Z(G) is the center of a group $C_{G}(\{x\})$ ...
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0answers
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If the tensor product of algebras $A \otimes B$ is unital, both $A$ and $B$ must be unital

It is clear that if $A$ and $B$ are unital algebras (over a field), then the tensor product $A \otimes B$ is also unital, with the unit being $1_A \otimes 1_B$. I came across an exercise that ...
2
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1answer
51 views

Kernel of a map in Graph Theory (toric ideals)

If we have an $n$-cycle with edges $e_1 =\{x_1,x_2 \}, e_2 = \{x_2, x_3 \},\dots, e_n = \{x_n,x_1\}$ with a $K-$algebra homomorphism $\phi: k[e_1,\dots, e_n] \to k[x_1,\dots, x_n]$ defined by ...
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0answers
23 views

Sylow counting to show group is isomorphic to semidirect product

Let G be a group of order $|G|=pq^m$ where $p, q$ are primes with $q^m<p$. Use a Sylow counting argument to show that $G\cong C_p \rtimes_h Q$ where $Q$ is a group with $|Q|=q^m$ and ...
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2answers
30 views

Show that a ring $R$ is a division ring if and only if, for each nonzero $a\in R$, there is a unique element $b\in R$ such that $aba = a$. [duplicate]

Show that a ring $R$ is a division ring if and only if, for each nonzero $a\in R$, there is a unique element $b\in R$ such that $aba = a$. $\Rightarrow$ Assume $R$ is a division ring. Let ...
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2answers
40 views

How to prove absence of a total order relation?

Show that on $\mathbb{C}$ (complex) there is no total order relation $≤$ such that both if the following properties hold $∀ (x, y, z) ∈ \mathbb{C}^3$, $x ≤ y \implies x + z ≤ y + z$ and $z ≥ 0, x ≤ ...
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0answers
30 views

How to show that a set of elements is a basis for the ring of integers of a number field?

Let $K$ be a number field of degree $n$ (that is $[K:\mathbb{Q}]=n$) with ring of integers $\mathcal{O}_K$. I know that there exists algorithms to find $\mathcal{O}_K$ and hence determine a ...
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1answer
30 views

Sylow's theorem for group of $2$ by $2$ matrices of determinant $1$ over the field of order $3$

Let $G=SL(2,\mathbb{F_3})$ - group of $2$ by $2$ matrices of determinant $1$ over the field of order $3$. (a) Find the order of $G$. I think it is $24$ but not sure how to verify it. (b) ...
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1answer
10 views

showing a function is surjective for isomorphisms

Consider a problem like the following. Prove that $ \mathbb{R} $ is isomorphic to the ring S of all $ 2 \times 2 $ matrices of the form $ \begin{bmatrix}a & 0\\0 & a\end{bmatrix}$, with $a ...
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0answers
30 views

Quick way to classify groups of small order

This is a past Qualifying exam on Algerbra : I'm curious if there is a quick way to solve Prob 1. To me, directly classifying groups takes too much time and I think I could not handle this in time ...
0
votes
1answer
65 views

Prove that these two fields are isomorphic.

I want to prove that $\bar{K}[V]/M_p \simeq \bar{K}$ where $K$ is a field, $\bar{K}$ is its algebraic closure and $$\bar{K}[V]=\bar{K}[x_1,...,x_n]/I_V,$$ where $I_V$ is the ideal attached to a ...
0
votes
2answers
20 views

Cosets of submodule

Suppose we have a ring $R$ and an ideal $J \subseteq R$. Let $M$ be an $R$-module and $N$ be a submodule of $M$. Assume for two elements $m$ and $m'$ of $M$ we have $$m+ JM=m'+JM.$$ Since $N ...
1
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1answer
27 views

Representation of a group and its quotient

Let $G$ be a (finite) group and let $N$ be a normal subgroup of $G$. Suppose that we have a representation $(V,\rho)$ of $N$ and a representation of $(V, \tau)$ of the quotient group $G/N$. Here $V$ ...
1
vote
2answers
65 views

Let $G$ be a group of order $35$. Prove that $G \cong C_5 \times C_7$.

Let $G$ be a group of order $35$. Prove that $G \cong C_5 \times C_7$. $G$ has subgroups of orders $5$ and $7$ by Lagrange's theorem? If so, call them $A$ and $B$. I know their intersection is ...
1
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2answers
21 views

Order of group $G = \{A\in M_2(\mathbb{Z}_p): \mathrm{det}A= \pm 1 \}$

Also, $p>2$ is a prime number. Firstly, it's obvious that $G \leq GL_2(\mathbb{Z}_p)$, and we know that $|GL_2(\mathbb{Z}_p)|=p(p^2-1)(p-1)$. Next, we define the homomorphic map ...