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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How to find automorphism of a particular order.

Given a finite group if the automorphism group is known is it possible to write down all the automorphisms with respective orders? For example say the group $Z_{p^{2}}$ has ...
1
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1answer
36 views

Confusion about the centre of a p-group

If a non-cyclic group $G$, non-commutative also has order $p^{3}$ does that mean for every $x\in G$ , $x^{p}$ is in $Z(G)$? I am trying to solve a problem from $p$-groups and at this point I ...
2
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0answers
12 views

Find the intertwiner of two equivalent group representations

Let $G$ be a finite group. Let $$\{A(g)\mid g \in G\} \text{ and }\ \ \ \{B(g) \mid g \in G\}$$ be two equivalent unitary matrix representations of $G$. How do I find a unitary intertwining matrix ...
0
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2answers
25 views

Field $F$ with $\operatorname{char}F=3$ and algebraic over $\mathbb{F}_3$ has a primitive root of unity.

Suppose that $F$ is a field with $\operatorname{char}F=3$ and $F$ is an algebraic extension of $\mathbb{F}_3$. Prove that $F$ contains a primitive $n$th root of unity for some $n>2$.
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33 views

Projective and injective resolutions of cyclic modules

Let $R$ be a ring and $M$ a cyclic $R$-module. It is well-known that always exist projective and injective resolutions of $M$. Is it any method to construct explicitly these resolutions which have the ...
1
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1answer
38 views

A polynomial's irreducibility in $\Bbb{Z}_p$

Show that if $f$ is irreducible in $\Bbb{Z}_p[x]$ then $f$ divides $x^{p^n} - x$ for some $n \in N$. I know that: $f$ is irreducible, so $F = \Bbb{Z}_p / {\left\langle f\right\rangle}$ is a ...
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2answers
59 views

Why do nil ideals annihilate simple modules?

A nil ideal $N$ of a ring $R$ is defined as follows: $(N,+)$ is a subgroup of $(R,+)$ $\forall x \in N, \forall r \in R :\quad x \cdot r \in N$ $\forall x \in N, \forall r \in R : \quad r \cdot x ...
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votes
1answer
43 views

a question of group theory [duplicate]

let $S$ be a collection of (isomorphism classes of) group $G$ which have the property that every element of $G$ commutes only with the identity element and itself then which option is true and why ? ...
2
votes
2answers
146 views

Group Theory: group under the composition multiplication modulo $p$

Suppose you have a group $G(S,*)$ where $S=\{1,2,\ldots,p-1\}$, $p$ is prime number, and $*$ is equivalent to the multiplication$\mod p$. If $a,b$ belong to $S$, then $ab\pmod{p}$ also belongs to ...
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0answers
20 views

$ G $ is supersoluble if and only if every normal subgroup of $ G $ with index no more than 2 satisfies the maximal permutizer condition.

Permutizer of a subgroup $ H $ of $ G $ is defined to be the subgroup generated by all cyclic subgroups of $ G $ that permute with $ H $, i.e. $ \langle x\in G \vert \langle x \rangle H = H \langle ...
2
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1answer
22 views

Decomposition of Torsion Module

Let $k$ be a field, $k[X]$ the polynomial ring in one variable and $M$ a torsion $k[X]$-module (not necessarily finitely generated). Consider the submodules \begin{equation*} M_1 = \{a \in M \mid ...
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2answers
27 views

Interpreting a Quotient Group ($D_8/\langle r^2\rangle$) in 2 Distinct Ways

I seem to have a misunderstanding about quotient groups. Let $D_{2n}$ denote the group of symmetries of an $n$-gon and let $V_4$ denote the Klein-4 group. On one hand, if we identify $r^2$ with $1$, ...
0
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1answer
36 views

The characteristic of real-closed fields is zero?

We know that $F$ is a real-closed field if $F$ is not algebraically closed but $F(\sqrt{-1})$ is algebraically closed. So I have this question What can we say about $\operatorname{char}F$? Is it ...
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1answer
52 views

Why the extension dimension of $x^3-2$ equal to $6$?

I have seen couple questions related to this one, but after reading the answers I am still confused: Why is the extension dimension of $x^3 - 2$ equal to $6$? In other words, why are the basis ...
2
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2answers
41 views

$K/F$ Galois Extension, $H \leq G$, where $G$ the Galois group, then there is $\alpha \in K$ s.t. $H=\{\sigma \in G : \sigma \alpha = \alpha\}$.

Let $K/F$ be a finite Galois extension of fields with Galois group $G$. Let $H$ be a subgroup of $G$. Then there is $\alpha \in K$ such that $H=\{\sigma \in G : \sigma \alpha = \alpha\}$. My proof ...
4
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3answers
123 views

Every normal subgroup is the kernel of some homomorphism

Clearly the kernel of a group homomorphism is normal, but I often hear my professor mention that any normal subgroup is the kernel of some homomorphism. This feels correct but isn't entirely obvious ...
2
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0answers
51 views

Composition factoring into nonnegative integer polynomials

Consider the integer polynomials with nonnegative coefficients, such as: $$ 1 + 2x +2x^2 $$ $$ 3 + 3x^4 + 11x^{10}$$ $$ 13 + 7x + x^2 $$ I asm interested in knowing "what is a composition ...
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1answer
48 views

True or false simple algebra questions (centralizers, conjugacy classes, normal groups, abelian groups) [closed]

Can someone please verify my answers to the following questions? Note: This is NOT homework! Answer true or false to the following questions: Two elements of a group in the same conjugacy ...
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0answers
30 views

Dihedral group and symmetry group of icosahedron

I am studying abstract algebra at the moment, but I have several troubles picturing the dihedral group $D_n$ and the symmetric group of the icosahedron. For instance, I find it really hard to solve ...
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1answer
30 views

Relating definitions of a normal field extension.

I have come across the following two definitions of a normal field extension. $\textbf{Definition 1:}$ An algebraic field extension $L/K$ is said to be normal if $L$ is the splitting field of a ...
0
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1answer
32 views

If $F$ is a field, then any two algebraic closures are isomorphic by an isomorphism that is the identity on $F$.

To start, suppose $K_1$ and $K_2$ are two algebraic closures of $F$. (a) Let $P$ be the set of partial functions $f$ from $K_1$ to $K_2$ with the following properties: $F$ is contained in ...
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1answer
20 views

Notation involving field extension

I am currently reading notes on Galois Theory and have come upon the following proposition, Let $f(x)$ be the minimal polynomial of a generator $\alpha$ of a finite field extension $k(\alpha)$ of ...
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22 views

$ G $ is satisfies the maximal permutizer condition if $ G $ is supersoluble?

Permutizer of a subgroup $ H $ of $ G $ is defined to be the subgroup generated by all cyclic subgroups of $ G $ that permute with $ H $, i.e. $ \langle x\in G \vert \langle x \rangle H = H \langle ...
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1answer
24 views

Semidirect Product of Two Groups

So I am beginning to learn Semidirect product. Now I have to identify the semidirect product of the two groups $Z_{p}\times Z_{p}$ =$H$ and $K$=$Z_{p}$ where p is an odd prime. So I can write ...
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3answers
395 views

Stanford math qual: Abelian groups $G$ satisfying $0\to \Bbb{Z} \oplus \Bbb{Z}/3\Bbb{Z} \to G \to \Bbb{Z} \oplus \Bbb{Z}/3\Bbb{Z} \to 0$

I am studying for my qualifying exams and came across the following question: Find all abelian groups $G$ that fit into an exact sequence $0\to \Bbb{Z} \oplus \Bbb{Z}/3\Bbb{Z} \to G \to ...
2
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1answer
52 views

Finding all ideals in a finite ring

Let $\mathbb F_2$ be the field of two elements. Consider the factor ring $$R=\mathbb F_2[x, y]/\langle x^2, y^2\rangle.$$ I want to find all ideals of $R$. Note that ...
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1answer
29 views

Equivalence of different prime factorizations in $\mathbb{Z}[\zeta_3]$

I'm reading that in the ring $\mathbb{Z}[\zeta_3]$, where $\zeta_3$ is the cubic root of unity, two prime factorizations of $4 = 2 \times 2 = (1 + \sqrt{-3})(1 - \sqrt{-3})$ are equivalent, because up ...
3
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1answer
65 views

Finding the Fixed Fields in the Galois Correspondence for the Splitting Field of $x^4-3x^2+4$ over $\mathbb{Q}$

I have found the Galois group of the polynomial $x^4 - 3x^2 + 4$ (see below), but I am not sure how to find the fixed fields in the Galois correspondence. The roots of the polynomial are $$\pm ...
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1answer
25 views

Prove socle is ideal

In any ring $R$ define the socle as the sum of all minimal right ideals of $R$. Say we have two minimal ideals $A,B$. If $a\in A,b\in B$, then $a+b$ is in the socle. If $x\in R$, then ...
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0answers
24 views

Question on primitive element for extensions of finite fields [duplicate]

I was given this question in field theory class which I am stumped on as after some effort I still cannot tackle. I am given a general natural number $n>1$ and am asked to prove or disprove: ...
2
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2answers
61 views

Ideal of polynomials vanishing on $\{(x,y): x^2+y^2=1, x \neq 0 \}$

I'm reading the book "Introduction to algebraic geometry" by Hassett, and in Chapter 3, after introducing the concept of the ideal of polynomials vanishing on a set $S$, the author gives some ...
2
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1answer
33 views

Another abstract algebra/field theory question

Suppose that $F$ is a field, $S \subseteq F^n$ and $I$ is an ideal in $F[x_1, \cdots, x_n] = F[\bar{x}]$. Define $$I(S) = \{ f \in F[\bar{x}]: f(\bar{s}) = 0, \forall \bar{s} \in S\}$$ and ...
2
votes
1answer
52 views

Number of abelian groups of order 108 [duplicate]

What is the number of abelian groups of order 108 upto isomorphism ? To answer this I wrote explicitly the possible abelian groups of order 108 as follows : $$\Bbb Z_{108}$$ $$\Bbb ...
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2answers
54 views

Cardinality of a ring obtained by quotienting $\Bbb Z[x]$

Let $R$ be the ring $\Bbb Z[x]/((x^2+x+1)(x^3 +x+1))$ and I be the ideal generated by $2$ in $R$. What is the cardinality of the ring $R/I$? 27 32 64 infinite Now I was thinking $R$ could be ...
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1answer
45 views

Infinite tower of algebraic extensions

For all I know, the following fact should be true: Consider an infinite tower of extensions $L_0 \subset L_1 \subset L_2 \subset \cdots$ such that $L_{i + 1} / L_i$ is algebraic for all $i \in ...
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3answers
28 views

Centre of matrix ring over skew field

Let $R$ be a semisimple ring. Show that $R$ is simple iff the centre of $R$ is a field. Book's solution: If $R$ is simple, it has the form $\mathfrak{M}_n(K)$ for a skew field $K$, and its ...
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1answer
42 views

Find the number of units in $M_2(\mathbb Z_p)$ where $p$ is prime. [duplicate]

Find the number of units in $M_2(\mathbb Z_p)$ where $p$ is prime. I want a detailed solution, not just the number. $M_2$ means matrix of order $2\times 2$. I know the defn of units. But how to ...
3
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0answers
35 views

Finding relations of variables

Suppose that \begin{align*} x&=t+t^{-1}+t^2s+t^{-2}s^{-1}+ts^{-1}+t^{-1}s-6\\ y&=t+t^{-2}+ts+s^{-1}-4\\ z&=t^{-1}+t^2+t^{-1}s^{-1}+s-4 \end{align*} Find a polynomial $P(x, y, z)=0$ ...
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0answers
74 views

What is the intuition behind Dirichlet's Class Number Formula? [closed]

As the title of the question suggests, what is the intuition behind Dirichlet's Class Number Formula being true? The Dirichlet Class Number Formula is$$h(\mathcal{O}_D) = -{1\over{D}} \sum_{n=1}^D ...
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0answers
60 views

Where does the term “Ring” come from in Algebra? [duplicate]

Group and Field make some sense to me, but I can't see why the structures that are closed under two binary operations would indicate "ring".
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0answers
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The space of alternating multilinear maps and existence of a bilinear map [duplicate]

Before, I ask similar this. But here I change question settings since it was incomplete. I hope receive good ideas. Let $E$ be a finite dimensional vector space over field $\mathbb R$ with $E^*$ as ...
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0answers
74 views

Show that $\mathbb{Z}_4\rightarrow \mathbb{Z}_8\oplus \mathbb{Z}_2\rightarrow \mathbb{Z}_4$ is exact [duplicate]

I want to know whether $0\rightarrow \mathbb{Z}_4\stackrel{f}\rightarrow \mathbb{Z}_8\oplus \mathbb{Z}_2\rightarrow \mathbb{Z}_4\rightarrow 0$ is exact wrt group homomorphism under addition. Since ...
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2answers
26 views

Group acting on $X$ and element of normal subgroup $H$ fixes an element of $X$ implies $H$ fixes all of $X$

A group $G$ acts on a set $X$ transitively and a normal subgroup $H$ fixes a point $x_{0} \in X$, i.e. $h \cdot x_{0}=x_{0}$ for all $h \in H$. Show that $h \cdot x = x$ for all $h \in H$ and $x \in ...
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0answers
29 views

Determinant of the change of basis for fractional ideal

Let $A$ be a fractional ideal of some number field extension $K:\Bbb Q$. Let $\omega_1, \dots ,\omega_n$ be a $\Bbb Z$ basis for $\mathcal O_K$ and let $\alpha_1, \dots ,\alpha_n$ be a $\Bbb Z$ basis ...
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4answers
83 views

Subgroup of $\mathbb{Z}$ generated by two positive integers

An exercise from Aluffi's Algebra book. Let $m,n$ be positive integers and consider the subgroup $\langle m,n\rangle$ of $\mathbb{Z}$ they generate. As a subgroup of $\mathbb{Z}$ it will be equal ...
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1answer
29 views

linear combination of polynomial equal to zero

I have a trouble with the following question. Let $p,q \in K[x]$. There are polynomials $a(x),b(X) \in K[x]$ with $\deg(a) < \deg(q)$ and $\deg(b) < \deg(p)$ such that $$a(x)p(x) + b(x)q(x) = ...
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2answers
57 views

If $G$ is a group with order $99$, it is cyclic by Sylow (isn't it?). I want to find a generator.

I have seen an argument in a specific case where $g,h\in G$ with $ord(g)=9$ and $ord(h)=11$ are used to create a generator through $f:=g^xh^y$ where $x, y \in \mathbb{Z}$ with $1=x9+y11$. Is this ...
2
votes
3answers
146 views

Find what $A/S$ is isomorphic to, where $A = \Bbb Z_2[X_1,\dots,X_n]$ and $S=\langle X_1^2-X_1,\dots,X_n^2-X_n\rangle$

Find what $A/S$ is isomorphic to, where $A = \Bbb Z_2[X_1,\dots,X_n]$ and $S=\langle X_1^2-X_1,\dots,X_n^2-X_n\rangle$. I'm sorry but I don't have anything to add here. I've been trying it with ...
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0answers
35 views

One particular permutation of 5 elements

As for options 1. and 2. I have taken a few examples randomly and found them to be correct but could not generalize and I am clueless about options 3. and 4. The answer ...
0
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1answer
33 views

Duality and tensor product of the Lie algebra

I would like to know how to compute the tensor product of the matrices below and how to deal with duality of vector spaces. The vector space I concern is the Lie algebra $\mathscr{sl_2}$ with basis ...