# Questions tagged [abstract-algebra]

For questions about monoids, groups, rings, modules, fields, vector spaces, algebras over fields, various types of lattices, and other such algebraic objects. Associate with related tags like [group-theory], [ring-theory], [modules], etc as necessary. To clarify which topic of abstract algebra is most related to your question and help other users when searching.

78,433 questions
Filter by
Sorted by
Tagged with
24 views

### Proof the follow exercise of group theory [closed]

let $a \in G$ where $G$ is finite. If $a^n$ has $m$ conjugates and $a$ has $k$ conjugates, the m|k.
16 views

### How do i generate cubic coordinate-space traversal functions?

Goal- generation of a function f(x) such that for each x = 0 to infinite, f(x) outputs a point in a coordinate space ...
41 views

### Show that if $G$ is finite and $G$ acts on $X$ and it is transitive, then $1=\dfrac{1}{|G|}\sum_{g\in G}|\operatorname{Fix}(g)|$ [closed]

I know you have to relate the $\sum_{g\in G} |\operatorname{Fix}(g)|$ with $\sum_{x\in X} |\operatorname{Stab}(g)|$ and use the orbit theorem and stabilizer but I get very confused.
• 1
24 views

### Extension of a representation

My question is related with this Representation abelian subgroup of an abelian group. I want to show that X extends to a representation of G, which I think it's obvious from G being defined by its ...
• 27
63 views

• 3,194
1 vote
25 views

52 views

### Is $\mathbb{Z}[\sqrt[3]{2}]$ a UFD? [duplicate]

Is the quotient ring $\mathbb{Z}[X]/(X^3-2)$ a UFD? I don’t have any idea to find whether some ring is a UFD or not.
42 views

### Math Questions Basic level [closed]

Find number of integers n such that n+3 divides n^3-3 ?
50 views

### How to prove that Quaternion's algebra over isomorphic to Mat2(Z [duplicate]

My ideas: I tried to build an explicit isomorphism, but as I think it is only possible when p = 1 (mod 4), and for p = 1 (mod 4) it get it. In my second attempt, I tried to look at them as vector ...
• 19
43 views

### If the product topology on $\mathbb{R}^2$ is all sets of the form $(a,b)\times(c,d)\subseteq\mathbb{R}^2$ what is product topology on $\mathbb{R}^4$?

Probably the answer to this question is too obvious for it to be asked but I want to make sure my assumption is correct. Consider the real line $\mathbb{R}$ with the topology that has basis consisting ...
• 157
9 views

### If Sigma(xn) and Sigma(yn) are convergent, show that Sigma(xn +yn) is convergent.

this is from exercise 3.7 introduction to real analysis bartle fourth edition number 4 i already tried to proof the Xn+Yn as a convergent but somehow it doesnt turn to proofed and still got some stuck
1 vote
31 views

### Naming convention: Lie groups with finitely generated discrete part

Consider a Lie group $G$ and let $G_0$ be the connected component of the identity. Then $G_{\mathrm{dis.}}:=G/G_0$ is a discrete group. We are writing a (physics) article in which finite-dim. Lie ...
• 107
54 views

30 views

### Prove a nonempty set $G$ with an associative operation $\ast$ is a group iff the following equations are satisfied $y \ast g = h$ and $g \ast x = h$ [duplicate]

I'm currently working on an exercise and the body of the text for the exercise is as follows. I have a first draft of the proof but am missing some things and am unsure about some things as well so ...
1 vote
43 views

### Equivalent Definition of injective $R$-module using exact sequences [duplicate]

We call an $R$-module $D$ injective if one of the two equivalent conditions holds: If $0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0$ is an exact sequence of $R$-modules then the induced ...
27 views

### Prove that a normal subgroup $G$ of $S_4$ with $(12)\in S_4$ is equivalent to the entire group $S_4$ [duplicate]

Consider a normal subgroup $G$ of $S_4$. The simple transposition $(12)\in G$. Prove that $G=S_4$. I have already proved that the above case implies that $(12),(23),(34)\in G$. These are all the ...
• 319
33 views

### Questions in structure theorems for artinian rings [duplicate]

This theorem was covered in the lecture notes of commutative algebra from where I am studying and I am struck on it on some points. Statement: Every artinian ring is a finite product of artinian ...
• 1,344
36 views

### $A \subset B$ rings, if $\alpha \in B$ integral, then $A[\alpha]$ integral over $A$?

Notation is from Lang, beginning of Chapter VII. (Extension of Rings). Rings are commutative with identity. $\alpha\in B$ is integral over $A$ if subring $A[\alpha]$ is finitely generated $A$-module (...
• 11
40 views

### Showing that $x^3 - t$ is irreducible over $\mathbb{F}_3(t)$

I was reading the post Is $\mathbb{F}_3(t,t^{1/3})/\mathbb{F}_3(t)$ a normal extension? Is it separable? I do not understand, why we can use Eisenstein's criterion to show that $x^3 - t$ is ...
• 81
1 vote
52 views

### Uniqueness of the ternary Golay codes

In [Van Lint - Introduction to Coding Theory] the uniqueness of the binary Golay codes is shown quite easily. In essence, the proof boils down to the fact that there is only one 2-$(11,5,2)$-design up ...
• 6,997
35 views

### A is noetherian and every prime ideal of A is maximal then...

This proof was part of my lecture notes in Commutative algebra and I am having trouble following it. So, I am asking the question here. Statement: If A is noetherian and every prime ideal of A is ...
• 1,344
37 views

29 views

### Where I am wrong in mod $p$ irreducibility test?

Mod p irreducibility test : Let $p$ be a prime an suppose that $f(x) \in \mathbb Z[x]$ with $\deg f(x) \geq 1$. Let $f_1(x)$ be the polynomial in $\mathbb Z_p[x]$ obtained from $f(x)$ by reducing all ...
• 694
1 vote
20 views

### Coinvariants of coaction $a\otimes b \mapsto \sum{\sigma_i(a)}\otimes \sigma_i(b)\otimes \sigma_i^*$

I've been studying Hopf-Galois Theory and currently I'm trying to understand some examples by writing all the explanations step by step by myself. The example I'm interested now is the classical ...
• 3,430
45 views

### Factoring polynomials modulo 3

Let $f(x) = x^5 + 2x^2 + 2x + 2 \in\mathbb Z_3[x]$. Then the irreducible factorization of $f(x)$ is $(x^2 +1)(x^3+2x+2)$ even though it does not have a root in $\mathbb Z_3$. How did we find that ...
24 views

78 views

### Maximal algebraic independent in commutative algebra over field

In field extension, maximal algebraic independent elements in a set of generators (generate by means of fraction of generators) will also be maximal alebraic independent amoung all subsets. It is then ...
• 21
43 views

### find all the prime ideals of $\mathbb{Z}_8$ [duplicate]

find all the prime ideal of $\mathbb{Z}_8$? My attempt : Positive divisor of $8$ are $1, 2, 4$ and $8$ So the ideal in $\mathbb{Z}_8$ are $(1)=\mathbb{Z}_8$ $(2)=\{0,2,4,6\}$ $(4)=\{0,4\}$ \$(...
• 47