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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Understanding the connection of the roots of an irreducible polynomial and a basis for field extensions

Let $\alpha,\beta,\gamma \in E$ be the roots of an IRREDUCIBLE polynomial $p(x)\in Q[x]$ (where E/Q is an extension field. Can I use these roots to construct a basis for E over Q? Why?
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22 views

For finitely generated free abelian groups $A,B$ if there is an onto homomorphism $A \to B$, then $\operatorname{rank}(A) \geq \operatorname{rank}(B)$

$\newcommand{\rank}{\operatorname{rank}}$For two finitely generated, free abelian groups $A,B$ prove that if there is an onto homomorphism $A \rightarrow B$, then $\rank(A) \geq \rank(B)$ Assume that ...
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4 views

How to create a ring in MAGMA with relations?

I'm using MAGMA221 and would like to create a ring over $GF(2)$ with respect to a list of relations. Here's what I have so far: $\mathtt{Z:=GF(2);} \\\mathtt{P<x,y,z>:=PolynomialRing(Z,3);}$ ...
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1answer
9 views

Showing one-to-one and onto

$\alpha$: $\mathbb{Z} \times \mathbb{Z}^{+} \rightarrow \mathbb{Q}$ defined by $\alpha(n,m)=\frac{n}{m}$ Is this one to one? Is this onto? I know that if $\alpha$ is one to one I must show ...
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22 views

Prove that the center of a group with $385$ elements has an element of order 7. [duplicate]

Prove that the center of a group with $385$ elements has an element of order 7. By Cauchy's theorem I know that if I prove that $Z(G)$'s order is divisable by 7, we're done. So now I need to rule out ...
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0answers
12 views

When the fibers of $A \to A[T]/(h(T))$ are geometrically regular?

Let $A$ be an affine commutative noetherian domain over a characteristic zero field $K$, $T$ an indeterminate, $h=h(T) \in A[T]$ (not necessarily monic), $B=A[T]/(h)$, and assume that $(h)$ is a prime ...
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1answer
31 views

Empty set - group [duplicate]

I started a course of algebra this morning, and the teacher explained the structure of a group. He explicitly explained a group has to be empty. Someone can explain to me why it is a necessity for a ...
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48 views

Between complex numbers and quaternions?

Complex numbers are $a+ b i $; Quaternions are $a + b i + c j + d k $. So, do there exist numbers like $a + b i + c j$? Here $a$, $b$, $c$, $d$ are all real. Did Hamilton consider such a case?
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1answer
16 views

Let $V=\{i\in \mathbb{Z}: 0\leq i< 2^n\}$. Define vector addition and scalar multiplication on $V$ to turn it into a vector space over $GF(2)$.

Let $V=\{i\in \mathbb{Z}: 0\leq i< 2^n\}$ for some $n\in \mathbb{N}$. Define vector addition and scalar multiplication on $V$ in such a way as to turn it into a vector space over the field $GF(2)$. ...
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1answer
36 views

Solving problem of abstract algebra [on hold]

I have understand the answer of my previous question.Thank you Sir.There is another question which I cannot answer. The question is that if $n$ is not a multiple of 23 then the remainder when ...
2
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0answers
58 views

Solving an equation of abstract algebra $x^{22}=2 \mod 23$ [on hold]

I have an abstract algebra problem which I am unable to solve.The problem is, if $x^{22}=2 \mod 23$, then $x$ has how many solutions? Please explain me.
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1answer
19 views

Evolution operator always in $SU(n)$?

Think about the evolution operator $U$ in Quantum Mechanics for finite dimensional systems. Then this operator satisfies an equation $$U'(t) = -iHU(t)..$$ Here, I assume that $H$ is ...
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1answer
26 views

How to prove thar O(Ng) | O(g)

I have this exercize: $G$ is a group. $N\subset G$. Need to prove that: $$o\left(Ng\right)\mid o(g)$$ where $Ng\in G/N$. For now, without using the canonic homomorphism $\tau \left(g\right)=Ng$. ...
2
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2answers
157 views

Group Action as permutations

I'm trying to study on Group actions. the paper says(if I understand) that if I have Set $S$ and an action $\alpha$: $G \times S \rightarrow S$ . then the action may be viewed as permutation by $x ...
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1answer
33 views

Are subgroups of order $p^{n-1}$ maximal?

Let $G$ be finite p-group of order $p^n$, I know all maximal subgroups of order $p^{n-1}$ Is it right to say all subgroup of order $p^{n-1}$ are maximal subgroups? If not, what property $G$ must have ...
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16 views

Rings leading to AKS primality test

Given number n, define ring $R = \Bbb Z_n[x]/(x^r −1)$ for a carefully chosen number $r$ ($r$ is much smaller than $n$; of the order of square of the number of digits in $n$) An element of $R$ is a ...
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39 views

Group of an order 385

Let $G$ be a group of order $385$, proof that $Z(G)~~ (cent(g))$ contains object of order $7$. I used sylow theorm and realized that there are $1$ sylow-$11$ sub-group which is normal in $G$ and also ...
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17 views

Enumerating functions modulo action on both the domain and codomain.

Let $Hom(A,B)$ be the set of functions from a finite set $A$ to a finite set $B$ and let $G_A \leq S_A$, $G_B \leq S_B$ be a subgroups of the permutation groups of $A$ and $B$. For $f,g \in Hom(A,B)$ ...
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25 views

Factor rings $R/R$ and $R/0$

Let $R$ be a ring. I want to describe the factor rings $R/R$ and $R/0$. So $R/R = \{[r]| r+R, \forall r\in R \}$ and since $r+R=R$, we get $R/ R =\{[0]\}$. And for $R/0 = \{[r]| r+0,\forall r\in ...
2
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1answer
21 views

Clarification on wording of a problem in Hungerford's Algebra

I'm currently working on problem 11 in section 1.1 of Hungerford's graduate text, which is to show 5 conditions of a group are equivalent. However, I'm not exactly sure what the last condition means: ...
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18 views

Elements that aren't left zero divisors are invertible for certain group algebra

Let $G$ be a finite group and $F$ a finite field with co-prime orders. Show that in the group algebra $F[G]$, if $x \in F[G]$ is not a left zero divisor then it is invertible. Thoughts so far: By ...
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23 views

There exists an $f\in \Bbb F[x]$ such that $I=\{fg|g\in \Bbb F[x]\}$.

$I\trianglelefteq \Bbb F[x]$. I want to prove that there exists an $f\in \Bbb F[x]$ such that $I=\{fg|g\in \Bbb F[x]\}$. I guess this means that I am meant to show that we have closure from the ring ...
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1answer
50 views

Define multiplication as $(a,b) \boxdot (c,d)=(ac-bd(r^2+s^2), ad+bc+2rbd)$ where $a,b,c,d,r\in \mathbb{R}$ and $0\neq s\in \mathbb{R}$.

Define multiplication as $(a,b) \boxdot (c,d)=(ac-bd(r^2+s^2), ad+bc+2rbd)$ where $a,b,c,d,r\in \mathbb{R}$ and $0\neq s\in \mathbb{R}$. The multiplicative inverse is $(1,0)$. I need to show that ...
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Elementry of analysis [on hold]

I don't understand this question, could you please define the question for me and tell me what should I do? Consider a particular device capable of arithmetic computation. Consider the operation ...
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0answers
36 views

The number of all groups with n elements? [on hold]

Let $n$ be a positive integer number. How many groups of $n$ elements (which are not isomorphic) ?
11
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1answer
127 views

Does $f(x) \in \mathbb{Z}[x]$ irreducible, imply $f(2x)$ also irreducible?

It is well known that $f(x+a) \in \mathbb{Z}[x]$ is irreducible when $f(x)$ is irreducible. I was wondering whether $f(2x)$ and in general whether $f(nx)$ is irreducible as well. Though it is ...
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2answers
39 views

How does uniqueness of the additive inverse imply that $-(ax) = (-a)x$?

How does uniqueness of the additive inverse imply that $-(ax) = (-a)x$? In my title, I should be clear that the additive inverse should be unique. But how does this help? I dont even get why ...
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36 views

Left remainder when dividing by $x-b$

Give a polynomial $p(x) = a_0 + a_1 x + ... a_n x^n \in \mathcal R[x]$ ($\mathcal R $ is any ring with unity), the book says when dividing $p(x)$ by $x-b \quad (b\in \mathcal R)$, the left remainder ...
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1answer
63 views

For an ideal $I \subset \mathbb{C} [x_1, … , x_n]$ show an iff about finiteness

For an ideal $I \subset \mathbb{C} [x_1, ... , x_n]$ show that dim$_{\mathbb{C}}R/I$ is finite iff $I$ is contained in only finitely many maximal ideals. Thoughts so far: I'm not sure how to get ...
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Suppose $ S = HF $ for $ H \leq S $ and $ H \cap F = 1 $. If $ N \leq H $, show $ N \leq Z(S) $.

Let $ G $ is a soluble group and $ N $ minimal normal subgroup of $ G $. Let $ F = Fit(G) $. Suppose $ S = HF $ for $ H \leq S $ and $ H \cap F = 1 $. If $ N \leq H $, show $ N \leq Z(S) $. I show ...
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1answer
29 views

Linear algebra: generalize from characteristic $0$ a problem about polynomial coefficients.

Let $K$ be a field, and let $F$ be a subfield of $K$. Assume that $F$ is infinite. Let $p(x)$ be a polynomial in one variable with coefficients in $K$, and suppose that $p(a) \in F$ whenever $a \in ...
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1answer
18 views

Linear transformations for fixing the line $y = 0$

The professor says that the subgroup for "stabilizing" the line $y = 0$ is $$A = \begin{bmatrix} a & c \\ 0 & d \end{bmatrix}$$ because in order to fix the first basis vector, $b = ...
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On Weisner Method

I read in chapter 2 "Weisner Method" in the book "Obtaining Generating Functions" by Elna Browning McBride In Sec 5" The extended form of the group generated by B and C " I did not understand ...
5
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1answer
31 views

Classification of all conjugacy classes of $GL_2(\mathbb{R})$, $GL_2(\mathbb{Q})$.

Give a classification of all conjugacy classes in the following groups. $GL_2(\mathbb{R})$ $GL_2(\mathbb{Q})$ My progress so far. If the characteristic polynomial splits, the matrix ...
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1answer
13 views

prove unique minimal normal subgroup of soluble group by $ P_{G}(M) > M $

Let $ G $ be a soluble group. If $ P_{G}(M) = \langle y\in G | \langle y \rangle M = M\langle y \rangle \rangle > M $for any subgroup $ M $ of prime power index in $ G $, then every chief factor ...
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$\mathbb{C}\{X\}^\chi$ a $G$-stable subspace of $\mathbb{C}\{X\}$?

Let $\mathbb{F}$ be a finite field with $q$ elements, and let $G = SL_2(\mathbb{F})$. The group $G$ acts on the set $X := \mathbb{F}^2 \setminus \{0\}$, the complement of the origin. For any group ...
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2answers
34 views

$(a,b) \mathbin\# (c,d)=(a+c,b+d)$ and $(a,b) \mathbin\&(c,d)=(ac-bd(r^2+s^2), ad+bc+2rbd)$. Multiplicative inverse?

Let $r\in \mathbb{R}$ and let $0\neq s \in \mathbb{R}$. Define operations $\#$ and $\&$ on $\mathbb{R}$ x $\mathbb{R}$ by $(a,b) \mathbin\#(c,d)=(a+c,b+d)$ and $(a,b) ...
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A quick Galois Theory question

Let $\mathbb{F}_q$ be a finite field of order $q$ where $q$ is a prime power. For any $d \in \mathbb{N},$ we have an inclusion $\mathbb{F}_{q^d} \subseteq \overline{\mathbb{F}}_q.$ Both ...
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Find a polynomial equation satisfied by $\phi$

I'm solving the following exercise from my class notes: Let $A=k[x^2,y^2]$ and let $M=k[x^2,xy,y^2]$ ($k$ a field). Show that $M$ is an $A$-module. Define $ \phi :M \to M$ by $\phi(m)=xym$. Find ...
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Number of matrices $A \in M_n(\mathbb{F}_q)$ where $A^2 = 0$.

What is the number of matrices $A \in M_n(\mathbb{F}_q)$ for which $A^2 = 0$ (as a function of $n$ and $q$)?
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1answer
57 views

Is there a “unique factorization theorem” for finite groups?

Sometimes it is difficult for me to understand what a group seems like. For example, the dihedral group $D_5$ is easy to visualise when I think it of as a "product" of two cyclic groups $C_2$ and ...
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4answers
44 views

Prove that $R$ is an integral domain $\Leftrightarrow$ $R[x]$ is an integral domain

Here is an exercise(p.129, ex.1.15) from Algebra: Chapter 0 by P.Aluffi. Prove that $R[x]$ is an integral domain if and only if $R$ is an integral domain. The implication part makes no problems, ...
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Is $V$ a simple $\text{End}_kV$-module?

Let $V$ be a finite-dimensional vector space over $k$ and $A = \text{End}_k V$. Is $V$ a simple $A$-module?
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Injective hull of a simple module

Let $M$ be an indecomposable injective right module over a right Artinian ring $R$, so $M$ has exactly one associated prime ideal $P$ (Lectures on Modules and Rings, T.Y. Lam). Now, $R/P$ is a simple ...
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1answer
68 views

Show that $k[x,y]/(xy-1)$ is not isomorphic to a polynomial ring in one variable.

Let $R=k[x,y]$ be a polynomial ring ($k$, of course, is a field). Show that $R/(xy-1)$ is not isomorphic to a polynomial ring in one variable. I can see that the polynomial $x+y$ is in ...
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1answer
37 views

$ S_{4} $ has a sylow tower?

A group having a Sylow tower is a finite group that possesses a Sylow tower: a normal series such that the successive quotient groups of the normal series all have orders that are powers of primes, ...
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1answer
27 views

Prove $v,w\in Z_{p}\times Z_{p}$ is linearly independent when $p=2$ and dependent when $p=3$

I need to prove that $\{v=(6,9),w=(7,8)\}\in Z_{p}\times Z_{p}$ is linearly independent when $p=2$ and linearly dependent when $p=3$. The problem is my freshman algebra course did not cover rings and ...
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28 views

Largest algebra in a vector space

Let $V$ be a vector space and let $C$ be a subset of $V$ that is closed under a bilinear operator $\langle \cdot, \cdot \rangle: V \times V \rightarrow V$. Let $A \subset V$ be an algebra containing ...
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52 views

Irreducible polynomials over the field $\mathbb{Z}_3[x]/\left\langle x^2+1\right\rangle$ [on hold]

We wish to find all the irreducible polynomials over the field $\mathbb{Z}_3[x]/\left\langle x^2+1\right\rangle$. I came across this problem in my course on advanced algebra. I have little knowledge ...
0
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1answer
21 views

Determining a lower bound on the order of a group based on its presentation

I am reading Abstract Algebra book by Dummit and Foote (3-rd edition). On pages 26-27 they define a dihedral group: $D_{2n} = \langle r,s | r^n = s^2 = 1, rs = sr^{-1} \rangle$ The authors ...