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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1answer
41 views

$p$-groups have normal subgroups of each order [duplicate]

Suppose $|G|=p^n$. Then $G$ has a normal subgroup of order $p^m$ for every $0\le m\le n$. By induction. It is clearly true for $n=0$. Now suppose $k<n$ and $H_i$ is a normal subgroup of $G$ of ...
1
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3answers
37 views

Trivial intersection of quotient subgroups

Suppose that $H/P$ and $M/P$ are two subgroups of a group $G/P$ such that intersect trivially i.e. $(H/P) \cap (M/P) = \{1_{G/P}\}$ = $\{P\}$. Is it true that $H \cap M = P?$
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0answers
18 views

Valuation fields and linear disjointness

I've got a question about linear disjointness of extension fields. Here I refer to the definition that Wikipedia gives: Two algebras $A$, $B$ over a field $k$ inside some field extension $\Omega$ ...
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0answers
20 views

Algebraic K-theory: induced maps

Let $f:A \to B$ be a homomorphism of rings. One way to define algebraic $K$ theory ($K_0$ group) is to consider the Grothendieck group of isomorphism classes of all finitely generated projective ...
1
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1answer
44 views

Representation of elements in a field

Let $A$ and $B$ be integrals domains, such that $A$ is integral over $B$. Writing $K(A)$ for the field of fractions, suppose that $K(A)$ is generated over $K(B)$ by a single element in $A$, say ...
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1answer
25 views

Dimension of a direct sum of characters (example with $S_3$)

Here is the character table of $S_3$: I was wondering how one can determine the dimension of for example the sign character $sgn$. Could we get it from the character table? Also, if we define $A$ ...
-1
votes
1answer
60 views

Example of an ideal which is not principal in the ring $\mathbb{Z} [x]$ [duplicate]

Give an example of an ideal in the ring $\mathbb{Z} [x]$ is not principal. What kind of example would be the easiest?
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1answer
23 views

Why if $Z/m\oplus Z/n = Z/mn$ then $(n,m)=1$ [duplicate]

Prove that if $Z/m\oplus Z/n = Z/mn$ then $(n,m)=1$. I have proved the converse, but here there is something I am missing. Hints instead of full answers are appreciated. Thanks.
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0answers
23 views

If $H$ and $K$ are nilpotent normal subgroups then $C(HK)$ is non trivial

I know that this follows from the fact that $HK$ is nilpotent but maybe there is an easier way to proof this? I wanted to show that there is an Element in $HK$ that commutes with $H$ and $K$. I ...
4
votes
2answers
82 views

The rigid additive tensor category freely generated by an object

I've found myself to be absolutely mystified by something in Deligne and Milne's notes on Tannakian categories. Namely, on p. 16 they are showing that there is a rigid additive tensor category ...
0
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0answers
46 views

Forgetful functor applied to a module

I try to find a left adjoint to the forgetful functor $U: R-Mod \longrightarrow Ab$. I considered a functor $F:Ab \longrightarrow R-Mod$ defined by $F(G)=Hom(U(R),G)$. I'm not so sure that in this ...
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2answers
52 views

Extension of intersection of ideals

Let $f:A \rightarrow B$ be ring homomorphism and $\mathfrak{a}_1,\mathfrak{a_2}$ be ideals of $A$. Let $\mathfrak{a}^e$ denote the extension of an ideal $\mathfrak{a}$ of $A$ in $B$. An exercise shows ...
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1answer
38 views

Elements in a Field of size $27$

I constructed the Field $$F_3[x]/<1 + 2x + x^3>$$ as the question asked to construct a field of size $27$ and I understood everything up to this point. The solution then says the elements in ...
0
votes
1answer
33 views

Solving a linear system in $\mathbb{Z}_{12}$?

Let $\alpha \in \mathbb{Z}_{12}$. I need to solve the following system in $\mathbb{Z}_{12, +, \cdot}$ for every $\alpha$ : \begin{cases} 6x + 5y = 0 \\ 8x + y = \alpha \end{cases} I'm confused because ...
2
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0answers
16 views

The simplest reference for Wedderburn decomposition

Assume $k$ is a field and $A$ is a $k$-algebra then the Wedderburn decomposition says $A=A_{sep}\oplus Nil(A)$, $A_{sep}\rightarrow A_{red}$ via $a\mapsto \bar{a}:=a+Nil(A)$. Where $A_{sep}$ means ...
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2answers
14 views

Arithmetic with congruence classes

I need to compute the following expression in $\mathbb{Z}_{5, +, \cdot}$ : $$ [2]_5^4 - [4]_5^4 \cdot [3]_5^4 \cdot [2]_5^4 $$ I'm not sure what is the best way to do this. Should I determine all the ...
-1
votes
0answers
27 views

Polynomial Path [duplicate]

Let $x(t)$ and $y(t)$ be real polynomials in $t$. Show that there is always a polynomial relation $f(x,y)=0$. This question is taken from Artin, Algebra, Chapter 3 Vector Spaces. I have no idea how ...
1
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1answer
41 views

$GL_2(\Bbb C)$ acts on a certain set

Let $G:=GL_2(\Bbb C)$, $B$ and $T$ be the subgroup consisting of all upper triangular and diagonal matrices in $G$, respectively. Set $w:= \left( \begin{array}{cc} 0 & 1\\ 1 & 0 ...
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0answers
25 views

What does it mean for a representation to be one-dimensional?

For example, take the Heisenberg Lie Algebra with the following basis: $X=\begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}$ $Y=\begin{bmatrix} 0 ...
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0answers
11 views

On the question of the Galois group of some polynomial. [duplicate]

I want to ask you some question on the Galois group of some polynomial. Let $p_1<p_2<\cdots<p_n$ be distinct prime numbers. Let's consider $f(t)=(t^2-p_1)(t^2-p_2)\cdots(t^2-p_n)\in ...
0
votes
1answer
20 views

Lie algebra homomorphism and representation

I am solving a multiple part problem on Lie algebra representations. I have done the first three parts, but am stuck on part (iv) as follows: Define a linear map $\phi : \mathbb{g} \rightarrow ...
2
votes
1answer
48 views

Determining the number of subgroups of $\Bbb Z_{14} \oplus \Bbb Z_{6}$

I want to determine how many subgroups does the additive group $G:=\Bbb Z_{14} \oplus \Bbb Z_{6}$ have? There are many related posts in our site, for instances: here and there. However, it ...
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0answers
35 views

Existence of algebraically closed extensions

Just want to check everything is fine. Basically, the point is to first take some polynomial ring with some huge amount of variables so that each can become a suitable root when we project it in the ...
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0answers
35 views

Minimal presentation of a tensor product of modules

Let $(R,m)$ be a local commutative noetherian ring. Suppose I have two modules $M$ and $N$ over $R$ given in terms of minimal presentations $$ M = \operatorname{coker}A, $$ and $$ N = ...
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2answers
34 views

Are the statements about the free $R$-module correct? [closed]

Let $R$ be a commutative ring with unit. If $F$ is a free $R$-module with finite rank, does it hold that each set of its generators contains a basis and that each linearly independent set of ...
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0answers
49 views

Show that they are isomorphic

Let $R$ be a commutative ring with unit and $M$ be a $R$-module. I want to show that $\text{Hom}_R(R,M)\cong M$ as $R$-modules and $\text{Hom}_R(R,R)\cong R$ as rings. $$$$ I have done the ...
0
votes
1answer
20 views

Does this stand for $\text{Hom}_R(R/I, R/I)$ ?

Let $R$ be a commutative ring with unit and $M$ be a $R$-module. We have that $\text{Hom}_R(R,M)\cong M$ as $R$-modules and $\text{Hom}_R(R,R)\cong R$ as rings. Do we have then also that ...
2
votes
3answers
77 views

Show that $S_5$ does not have a quotient group isomorphic to $S_4$

Show that $S_5$ does not have a quotient group isomorphic to $S_4$. If we to assume that $H$ is such a group, than $H$ must be normal in $S_5$ and $|H|=|S_5|/|S_4|=5$. So $H$ must be isomorphic ...
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0answers
23 views

Dimension of induced representation in $S_3$

Let $G=S_3$. It has 3 irreducible representations: $1, sgn$ and $V$; the trivial rep, sign rep and rep $V$ where $dimV=2$ Consider the subgroup $H=S_2$ with irreps $1_H$ and $sgn_H$ What is the ...
2
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2answers
48 views

Quotient of $G= \left\langle a, b \ \middle|\ a^4, b^2=a^4, aba=b \right\rangle$ by $\langle a^2 \rangle$

Let $G$ be finite group of order $8$ of the form: $G= \left\langle a, b \ \middle|\ a^4, b^2=a^4, aba=b \right\rangle$. The elements are $\left\lbrace 1, a, a^2, a^3, b, ab, a^2b, ...
1
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1answer
39 views

Why is $\mathbb{F}_5[x]$ a Jacobson ring? [closed]

As the question title suggests, why is $\mathbb{F}_5[x]$ a Jacobson ring?
2
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1answer
30 views

The Galois group of a polynomial over a field and over some extension (updated)

Let $f(x)\in F[ x]$ and $K/F$ be a field extension. Show that the Galois group of $f(x)$ over $K$ is isomorphic to a subgroup of the Galois group of $f(x)$ over $F$. Let $E$ be the splitting ...
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0answers
21 views

Row in the character table of $D_{10}$

Give the values of one row of the character table of $D_{10}$ corresponding to a character of degree $2$ I know the conjugacy classes of $D_{10}$, the dimensions of the irreducible ...
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1answer
24 views

Character of a representation on $S_3$ and irreducible representations

Here is the character table of S3: Consider $V=\mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^2$ with basis $e_{ijk} := e_i \otimes e_j \otimes e_k $ Let $\pi$ be the representation of ...
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0answers
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Defining a filtration on free Lie algebra in terms of generators

Let $L$ be the free Lie algebra generated by the set $X = \{x_i\}_{i \in \mathbb{N}}$ over a field $k$. A Lie monomial is a bracketed word of elements of $X$ of finite length, and $L$ is spanned by ...
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0answers
54 views

What does the irreducible polynomial over $\mathbb{Q}$ and $\mathbb{Q}(i)$ mean?

So here'as the problem Find the monic irreducible polynomial $g(x) \in \mathbb{Q}(x)$ for $i+ \sqrt{3}$ over $\mathbb{Q}(i)$ Erm...huh? Okay, so a minimal polynomial $m$ is what I need, and ...
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0answers
24 views

Good abstract algebra textbook [duplicate]

I want to get a head start for an abstract algebra class I'm taking next year. Are there any textbooks that are highly recommended?
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3answers
40 views

Is it true that a polynomial is reducible over a field only if the polynomial has a zero in the field?

I am doing some practice problems for abstract algebra and have come across this idea in a couple places, but it seems fundamentally wrong. For example, in order to prove that $f(x) = x^2 + x + 1$ is ...
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0answers
44 views

Categorical Representation of the Set of All Strings

Let $A$ be a small preordered category. How would we define a preordered category $\mathcal A$ for all strings over $A$ (e.g., Kleene Closure) ordered by the subword order (Def'n $3.1$) $\leqslant$? ...
2
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3answers
35 views

Why number of bases of $\mathbb{F}_p^2$ equals order of $GL_2(\mathbb{F}_p)$?

Artin, Algebra, Chapter 3, Ex. 4.4 I can prove (b), viz., that The order of $GL_2(\mathbb{F}_p)=p(p+1)(p-1)^2$ The order of $SL_2(\mathbb{F}_p)=p(p+1)(p-1)$ However, I have no idea how to prove ...
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0answers
24 views

Does there exist a group G such that G is countable but collection of subgroups of G is uncountable? [duplicate]

Does there exist a group G such that G is countable but collection of subgroups of G is uncountable? I have tried with $\mathbb{Z}, \mathbb{Q}$ But I could not able to end up and later I choose ...
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0answers
20 views

Maximal ideal of subalgebra over a field [duplicate]

Let $A$ a finite $k$-algebra (with $k$ a field) and $B$ a subalgebra of $A$. Prove that if $\mathfrak{m}$ is a maximal ideal of $A$ then $\mathfrak{m}\cap B$ is a maximal ideal of $B$. It is easy ...
3
votes
1answer
43 views

Every group element is a product of elements in certain subsets

Let $G$ be a group. For $\theta \in$ Aut$(G)$ of order $2$, define $$ K:=\{ g\in G \mid \theta(g)=g \},\quad S:=\{ \theta(g)^{-1}g \mid g\in G \}.$$ My first question is: Assume there is a ...
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0answers
38 views

How to understand the formula $\Delta(x_1 \otimes x_2) = \Delta(x_1)\Delta(x_2)$?

In the webpage, it is said that the comultiplication on the tensor algebra $TV$ is defined as follows. \begin{align} \Delta(x_1\otimes\dots\otimes x_m) = \sum_{p=0}^m ...
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1answer
49 views

Question about finite abelian group

Let G be an abelian group of order $mn$ where $\gcd(m,n)=1$. I proved that $mG$ and $nG$ are subgroups and that $G=mG+nG$ and now i want to prove the three things: the sum is direct, i.e. $mG\cap ...
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2answers
49 views

How to calculate permutation $(12)^{-1}(12345)(12)$ [closed]

I was wondering if someone could help me find $(12)^{-1}(12345)(12)$ I need to know this for calculating conjucacy classes and then a character table, thanks
1
vote
1answer
34 views

Why commutator subgroup is normal to G? [duplicate]

If $G$ is a group and $G'$ is generated by $\{xyx^{-1}y^{-1}|x,y\in G\}$, then $G'\trianglelefteq G$ and $G/G'$ is Abelian. At first, I thought this is easy because I thought ...
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2answers
54 views

If every intermediate ring of a field extension is a field, then the extension is algebraic

Suppose $E/F$ is an extension of fields. Prove that if every ring $R$ with $F\subseteq R\subseteq E$ is a field, then $E/F$ is an algebraic extension. I can show the converse is true by ...
0
votes
1answer
24 views

Module over an algebra, what is it?

Let $A$ an algebra of the field $\mathbb F$. What is an $A-$module. I was thinking that $B$ is an $A-$module if 1) $A(+,\cdot ,*)$ is a $\mathbb F-$vector space 2) $*:B\times B\longrightarrow B$ is ...
4
votes
1answer
52 views

If two matrices are path connected, so are their inverses

The set of $n\times n$ matrices can be identified with the space $\mathbb{R}^{n\times n}$. Let $G \le GL_n(\mathbb{R})$. We say that $A \in G$ and $B \in G$ are path-connected (not sure if this is ...