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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-3
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1answer
50 views

Isomorphic or not: two infinite groups [on hold]

The groups $(\mathbb{C}\setminus \left \{ 0 \right \},\cdot )$ and $(\mathbb{R},+)$ are not isomorphic. So I was not clear whether this statement is true or not.
3
votes
0answers
37 views

Kähler differentials in an inseparable field extension

Let $L/K$ be a finite (or, more generally, algebraic) field extension. It is easy to show that if $L/K$ is separable then the $L$-vector space $\Omega_{L/K}$ of relative Kähler differentials is zero. ...
0
votes
2answers
46 views

General questions about Polynomial Rings [on hold]

I'm learning about polynomial rings in my class. My instructor and book are both spectacularly unhelpful and didn't even bother to define most of the terms in my homework. So I have some general ...
1
vote
1answer
20 views

Let $K$ be a field and $f(x)\in K[X]$ be a polynomial of degree $n$. And let $F$ be its decomposition field. Show that $[F:K]$ divides $n!$.

Let $K$ be a field and $f(x)\in K[X]$ be a polynomial of degree $n$. And let $F$ be its decomposition field. Show that $[F:K]$ divides $n!$. Here $[F:K]$ denotes the dimention of $F$ over $K$ as a ...
1
vote
1answer
25 views

$f(x) \in\Bbb Q[x]$. Prove that if $f(a + b\sqrt c) = 0$, where $a, b \in\Bbb Q$ and $\sqrt c \in\Bbb Q$ then $f(a − b\sqrt c) = 0$. [on hold]

Let $f(x)\in\Bbb Q[x]$. Prove that if $f(a + b\sqrt c) = 0$, where $a, b \in\Bbb Q$ and $\sqrt c \not\in \Bbb Q$ then $f(a − b\sqrt c) = 0$. I don't really have any idea of where to start on this. ...
3
votes
1answer
35 views

$\{(1), (12)(34), (13)(24), (14)(23)\}$ is the only non-cyclic proper subgroup of $A_4$?

What can I use to display the following: $\{(1), (12)(34), (13)(24), (14)(23)\}$ is the only non-cyclic proper subgroup of $A_4$. What I've started to do: list all the elements of $A_4$ and finding ...
4
votes
1answer
60 views

Prove that if Aut($G$) is the trivial group, then so is $G$? [duplicate]

Let $G$ be a finitely generated group. Show that if Aut($G$) is the trivial group, then so is $G$. I know that if Aut($G$) is the trivial group then $G$ must be abelian but I'm not sure how to ...
1
vote
1answer
22 views

Let R* be the set of units of R and S* be the set of units of S. Prove that f(R*) = S*.

Let R and S be commutative rings with unity $1_R$ and $1_S$ respectively, and let $f: R\to S$ be a ring isomorphism. I am at a loss. Any help is much appreciated.
0
votes
1answer
66 views

What is an order of an element of a partition"?

I'm reading a paper, in which the set of all 3^3 mappings from {0,1,2} to itself (for instance {001,020,110,121,122}, {002,010,112,011}, {0,1,2}, ...) is partitioned, after which is written two ...
-1
votes
0answers
21 views

Subgroup $K$ of $G$ and $a \in G$ $\implies$ $Ka=\{ka: k \in K\}$? [on hold]

I'm asked to prove the following: Let $K$ be subgroup of group $G$ and $a \in G$. Show that $Ka=\{ka: k \in K\}$. But isn't this the definition of a right coset? What do I need to prove?
4
votes
1answer
36 views

Constructing well-defined epimorphism onto $S_3$?

Let $\Lambda$ denote the group with presentation $\langle a,b \mid abab^{-1}a^{-1}b^{-1}\rangle$. Construct an epimorphism from $\Lambda$ onto $S_3$, making sure to check that the function is ...
3
votes
3answers
57 views

Show that the extension $F \vert \mathbb{Q}$ is Galois

Let $E \subset \mathbb{R}$ a field containing $\mathbb{Q}$ such that the extension $E \vert \mathbb{Q}$ is Galois, and let $F := E(\sqrt{-1})$. Show that the extension $F \vert \mathbb{Q}$ is ...
0
votes
0answers
17 views

$U$ subset of $S$, $G$ acts transitively on $S$, show that the subsets $gU$ cover $S$ evenly

I have a finite set $S$ on which a group $G$ acts transitively. Now, I let $U$ be a subset of $S$. I want to show that the subsets $gU$ cover $S$ evenly, meaning every element of $S$ is in the same ...
1
vote
0answers
41 views

Prove that $\sin ^{-1} 1 $ is algebraic over $\mathbb Q$

Prove or disprove the following : $1.\sin ^{-1} 1 $ is algebraic over $\mathbb Q$ $2.\cos (\frac{\pi}{17})$ is algebraic over $\mathbb Q$ As suggested by @Andre ,for the 2nd one ...
0
votes
1answer
21 views

Polynomial ring indexed by an arbitrary set.

Let $B$ be any non-empty set, possibly uncountable. What does the term a polynomial ring indexed by the set $B$ means?
0
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2answers
22 views

Homomorphism of groups

Let $\phi : G_1 \rightarrow G_2$ be a homomoprhism of groups. Now I have to prove that for any $g \in G_1$ we have $\phi (g^{-1})=[\phi (g)]^{-1}$ So how should I begin?
0
votes
2answers
31 views

$f(x) \in \mathbb{R}[x]$. Prove that if $z = a + bi$ is a zero of $f(x)$ then $z = a − bi$ is also a zero of $f(x)$.

Let $f(x) \in \mathbb{R}[x]$. Prove that if $z = a + bi$ is a zero of $f(x)$ then $z = a − bi$ is also a zero of $f(x)$. I'm learning about polynomial rings but my book and my instructor never ...
0
votes
2answers
19 views

Group action decomposes $X$ into distinct orbits

Define the group action as $g\cdot x:=g^{-1}xg.$ Let $G=A_5$, and $X=\{\sigma\in A_5:=\sigma=(a,b,c,d,e)\}.$ Show that the group action on X decomposes $X$ into two distinct orbits. There are 60 ...
1
vote
1answer
20 views

Representation/Character theory of $S_3$: What is the Vector space $V$?

This is a basic question that I may have a misunderstanding on. When we study the character table of a group, say $S_3$, what vector space are we looking at? I understand that a linear ...
0
votes
3answers
30 views

Why isn't the number of cosets equal to cardinalities of the groups?

The left coset for subgroup $H$ of $G$ and element $x \in G$ is $$xH=\{xh : h \in H\}$$ Now why is the number of left (or right) cosets not equal to $|G||H|$? Since if one picks $x \in G$ for every ...
6
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2answers
62 views
+50

$R$ be a commutative ring with unity such that every surjective ring homomorphism $f:R\to R$ is injective , then is $R$ Noetherian?

Let $R$ be a commutative ring with unity such that every surjective ring homomorphism $f:R\to R$ is injective , then is $R$ a Noetherian ring ?
0
votes
1answer
27 views

Prove that if $J \subset S$ is a principal ideal of $S$, then $f^{-1}(J)$ is a principal ideal of $R$.

Let $R$ and $S$ be commutative rings with unity $1_R$ and $1_S$ respectively, and let $f: R \to S$ be a ring isomorphism . Prove that if $J \subset S$ is a principal ideal of $S$, then $f^{-1}(J)$ is ...
2
votes
1answer
32 views

Confusion with Closures in the topological sense

The rigorous definition is A closure of $A \subseteq X$ of a topological space $X$ is denoted Cl($A$) and is the intersection of all closed subsets of $X$ that contain $A$. The more intuitive or ...
4
votes
0answers
43 views

Is my proof that $ \mathbb{Q}(\sqrt{5}) $ is the only quadratic subfield of $ \mathbb{Q}(\zeta_5) $ correct?

I would like some feedback on my proof of the fact stated in the question. First, we note that $ L = \mathbb{Q}(\zeta_5) $ is the splitting field of $ X^5 - 1 $, so it is Galois. Furthermore, we know ...
0
votes
1answer
54 views

Don't understand the proof of Artin's “Algebra” Ed 1, Prop 5-8.4

I'm reading Artin's Algebra, Edition 1. In Chapter 5 there's proposition (8.4): Let $c_g$ denote conjugation by $g$, the map $c_g(x) = gxg^{-1}$. The map $f: S_3 \rightarrow Aut(S_3)$ from the ...
0
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0answers
23 views

Centralizer of $\sigma\in S_n$ [duplicate]

Let $\sigma\in S_n$ . Describe the centralizer of $\sigma$. Thought: If I conjugate $\sigma$, then $\tau^{-1}\sigma\tau=\sigma.$ This means that $\tau$ is a power of $\sigma,$ and ...
1
vote
1answer
44 views

If $x+y\sqrt{n} \in \mathbb{C}$ is a root of $f$ then $x-y\sqrt{n}$ is also a root

Let $n\in \mathbb{Z}$ be a non-square integer and $x+y\sqrt{n} \in \mathbb{C}$ a root of $f\in \mathbb{Q}[x]$ with $x,y\in \mathbb{Q}$. Show that $x-y\sqrt{n}$ is also a root of $f$. To show ...
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0answers
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Dimension of Algebraic Variety

What is the dimension of the algebraic variety formed by $2m \times n$ real matrices, where all $(r_1+1) \times (r_1+1)$ minors of the top $m \times n$ matrix vanish, all $(r_2+1) \times (r_2+1)$ ...
16
votes
2answers
148 views

Is there a fundamental theorem of algebra for matrices?

The fundamental theorem of algebra says we can do this ($z\in\mathbb{C}$ of course) $$\sum_{k=0}^n a_kz^k= a_n\prod_{k=1}^n (z-\omega_k)=0$$ for some set $\{\omega_k \in\mathbb{C}\}_{k=1,2,\ldots , ...
-1
votes
0answers
38 views

$A,B\in\mathbb Q[x]$ with $A,B$ monic, and $ AB\in\mathbb Z[x]$, prove $A,B\in\mathbb Z[x]$

It is part of cyclotomic polynomials. But I don't know how to deal with it and what to do next. I have prove $n$-th root is related to Euler's totient fuction. But I don't know how to use it. Thank ...
9
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1answer
79 views

Commutativity of a ring from idempotents.

In a ring $R$ with unity, every element can be written as product of finitely many idempotents. Can one show that the ring is commutative?
2
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2answers
44 views

Proving that a field is not a splitting field of any polynomial

I am trying to prove that $Q(2^{1/3})$ is not a splitting field of any polynomial $p(x) \in Q[x]$. I know I need to show that there does not exist any polynomial in $Q[x]$ that has all of its roots ...
3
votes
0answers
28 views

Tensor Product of Modules over R and a subring S

If we have a commutative Ring R with a subring S and two R-Modules (or S-Modules). What can we say about the correlations between $$ M \otimes_R N \quad and \quad M \otimes_S N \quad ?$$ Wikipedia ...
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0answers
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Example of lattice of subgroups of quotient group [on hold]

I've studied a theorem that explains what is the lattice of subgroups of a quotient group. The result is the following: Given a group G and a normal subgroup N if we denote by Sub(G) the lattice ...
1
vote
2answers
28 views

How to show that $X^p-t\in\mathbb{F}_p(t)[x]$ is irreducible? [duplicate]

This question is previously asked here, but there is no complete solution of it. I understand that the root $\alpha$ exist in the algebraic closure of $\mathbb{F}_p(t)[x]$, and it is the only root ...
-3
votes
1answer
22 views

The mapping defines a unique automorphism [on hold]

Let $R$ be a commutative ring with unity and $a,b\in R$ with $a$ invertible. I want to show that the mapping $x\rightarrow ax+b$ defines a unique automorphism of $R[x]$ that is idempotent in $R$. ...
3
votes
1answer
31 views

Show the intersection of six subgroups of order $24$ is normal in $G$? [duplicate]

Let $G$ be a group with exactly six subgroups of order $24$. Show that the intersection of these six subgroups is normal in $G$. My thought is that if we can show these six subgroups are normal ...
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0answers
40 views

I want to find questions or problems to do math research. [on hold]

This is a soft question, but I am not sure of a better forum. I am looking for a professor who can help me find a question or problem that I can independently research and try to solve in order to ...
1
vote
1answer
60 views

Do we have to show that $f(x)\in R$?

Let $R$ be a commutative ring with unity. I want to show that if $g(x)=c_nx^n+\dots+c_0\in R[x]$ is a zero divisor of $R[x]$ then there exists $d\in R \setminus \{0\}$ such that $dc_n=dc_{n-1}=\dots ...
2
votes
1answer
49 views

Show that if a group $G$ has odd order, any subgroup $H$ of index 3 in $G$ is normal in $G$? [duplicate]

Show that if a group $G$ has odd order, any subgroup $H$ of index 3 in $G$ is normal in $G$. I think this is equivalent to the following: Let $H$ and $K$ be subgroups of a group $G$, with $K ...
5
votes
1answer
43 views

Group with exactly six Sylow 5-subgroups?

Give an example of a group with exactly six Sylow 5-subgroups. I think $A_5$ works because it has 6 subgroups of order 5: $\langle(12345)\rangle,\langle(12354)\rangle, \langle(12435)\rangle, ...
5
votes
1answer
68 views

Example of a non-abelian quotient of a non-abelian finite group?

Give an example of a non-abelian quotient of a non-abelian finite group. This should be fairly simple but I am drawing a blank. I can think of plenty of non-abelian, finite groups but no ...
1
vote
3answers
26 views

Simple-to-play morra game to select m winners from n contestants.

I have m apples and n people (m < n) and we need to play a fair deterministic game to decide who gets the apples. I know how to do this if m is 1 with morra, having each player submit an integer ...
0
votes
1answer
17 views

Example of a ring R such that characteristic(R) = 0, R is not a field, and R has at least 4 units elements [on hold]

I am at a loss. Any help is greatly appreciated. I assume R has to be infinite.
1
vote
1answer
43 views

For which $n$ in $\mathbb Z/n\mathbb Z$, $\bar{2}$ has a multiplicative inverse.

I am looking for which $n$ in $\mathbb Z/n\mathbb Z$, $\bar{2}$ has a multiplicative inverse. Attempt: I know that I need a $\bar{k}$ such that $\bar{k}$$\bar{2}$ $= \bar{1}$. I believe that the ...
7
votes
2answers
50 views

With the exception of $\mathbb{Z}$, every infinite abelian group contains a subgroup isomorphic to $\mathbb{Z}^2$?

With the exception of $\mathbb{Z}$, every infinite abelian group contains a subgroup isomorphic to $\mathbb{Z}^2$. Is this statement true? I don't have much experience working with non-finite ...
5
votes
1answer
37 views

The centralizer of $x_3x_2x_1x_2x_1x_3^{-1}$ in $F_3$ is $\langle x_3x_2x_1x_2x_1x_3^{-1}\rangle$?

The centralizer of $x_3x_2x_1x_2x_1x_3^{-1}$ in $F_3$ is $\langle x_3x_2x_1x_2x_1x_3^{-1}\rangle$. The centralizer of an element in $F_3$ is the set of elements of $F_3$ that commute with that ...
5
votes
4answers
52 views

Let $H$ and $K$ be subgroups of $G$ such that $H \cap K = \{e\}$. Then $H \cup K$ is a subgroup of $G$?

Let $H$ and $K$ be subgroups of $G$ such that $H \cap K = \{e\}$. Then $H \cup K$ is a subgroup of $G$. I know that $H \cup K$ is a subgroup of $G$ if and only if $H \subseteq K$ or $K ...
7
votes
0answers
56 views

A List of Standard or “Cliche” Homeomorphisms [duplicate]

Learning topology has been hard. I just cannot see how some people can come up with complex functions that link one space to another, in a homeomorphic sense. The explanations are always "Well if you ...
3
votes
0answers
31 views

Computing center of an algebra

Let us define an associative algebra over $\mathbb{C}$ with generators $x, y, z$ and the following relations: $x^2=x, y^2=y, z^2=z, 2yxy=y, 3zyz=z, xz=zx$. I am interested in finding center of this ...