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Questions tagged [abstract-algebra]

For questions about groups, rings, fields, vector spaces, modules and other algebraic objects. Associate with related tags like group-theory, ring-theory, modules, etc. to clarify which topic of abstract algebra is most related to your question and help other users when searching.

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10 views

The multiplicative group of units in $\mathbb{Z}_{p}$ is isomorphic to $\mathbb{Z}_{p} \times C_{n}$ with $n=\max\{p-1,2\}$.

Problem. (a) Write down explicitly the rules for addition and multiplication in $\mathbb{Z}_{p}$. (b) Show that $(\mathbb{Z}_{p^{i}},\varphi_{ij})$ where $\varphi:\mathbb{Z}_{p^{j}} \to \mathbb{...
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0answers
46 views

How to proceed with following problem of algebra

How to proceed with this problem. I tried to form equation but it cant be solve with that. $$\begin{align}\dfrac{a^2}{2^2 - 1^2} + \dfrac{b^2}{4^2 - 1^2} + \dfrac{c^2}{6^2 - 1^2} + \dfrac{d^2}{8^2 - ...
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0answers
8 views

Generator matrix of Reed-Solomon code

Given the [10, 3, 8] Reed-Solomon code over $F_{11}$ with evaluation points $\alpha_i = i$ for $i = 1, 2, ..., 10$. How can I construct, with these information, the generator matrix?
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0answers
7 views

Inverse limit of projective profinite groups is projective

I'm trying to prove the following (H.W question): Let $ G $ be inverse limit of projective profinite groups. Prove that $G$ is projective group. Projective group means "cohomological dimension ...
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0answers
8 views

Generator matrix for binary Goppa code

Let $F_8 = F_2 (\alpha)$, where $\alpha$ is a root of $x^3+x+1$. I'd like to construct the generator matrix for a binary Goppa code $T(L, G)$ with Goppa polynomial $G(x) = x+1 \in F_8[x]$ and L the ...
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0answers
13 views

Surjectivity of a bilinear map between groups

Let $G$ a generic group and $\gamma_i(G)$ its lower central series defined as $$ \begin{cases} \gamma_1(G)=G\\ \gamma_{i+1}(G)=\left[\gamma_i(G); G\right] \end{cases} $$ for every $i\in \mathbb N$. ...
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1answer
23 views

How to show that some elements of $\mathbb{Z}[2\sqrt{2}]$ are irreducible?

I want to show that $2$ and $2\sqrt{2}$ are irreducible in $\mathbb{Z}[2\sqrt{2}]$. Consider the norm $N:\mathbb{Z}[2\sqrt{2}]\to\mathbb{Z}_{\ge0}$ defined by $N(a+b\cdot2\sqrt{2})=a^{2}-8b^{2}$. ...
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0answers
8 views

Effect of puncturing and shortening codes

Let $C$ be a linear [n, k, d] code. Then, $\bar{C} = [n-1, \bar{k}, \bar{d}]$ is an i-punctured code of C. (The i-th coordinate is deleted.) And $C' = [n-1, k', d']$ is an i-shortened code of C (...
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1answer
23 views

Help understanding a theorem about group homology $H_0$

I'm self-studying homological algebra. I have problems on understanding a theorem about $H_0$. First, I don't know where the bottom row of the commutative diagram comes from, see the red line in ...
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14 views

Minimal projective presentations from projective presentations.

Let $M$ lie in $\operatorname{mod}\Lambda$ for a finite dimensional $k$-algebra $\Lambda$. Let $P_1 \xrightarrow{d} P_{0} \xrightarrow{\pi} M$ be a projective presentation of $M$. Then, one has a ...
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0answers
21 views

Inner product structure on geometric algebra?

I understand that geometric algebra equips itself with the contraction operators $\rfloor$ and $\lfloor$. While these are awesome when one wishes to project a subspace onto another, it is not an inner ...
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0answers
20 views

Reducibility Unnecessary Hypothesis?

Problem: Let $F$ be a field with $\text{char} F = p$ for some prime $p$. Show that if $X^p - X - a$ is reducible in $F[X]$, then it splits into distinct factors in $F[X]$ Solution in back of the ...
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1answer
27 views

$M$ is cyclic if and only if there is some ideal $I\subset R$ such that $R/I\cong M$

This is the first part of exercise 11 on page 168 of Advanced linear algebra third edition of Steven Roman Let $M$ an $R$-module, then prove that $M$ is cyclic if and only if there is some ideal $I\...
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1answer
34 views

Canonical ring structure on the tensor product $R \otimes_\mathbf{Z} S$.

Let $R$ and $S$ be commutative rings. I need to show that there is a unique ring structure on the $\mathbf{Z}$-module $T := R \otimes_\mathbf{Z} S$ such that $$ (r_1 \otimes s_1)(r_2 \otimes s_2) = ...
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1answer
30 views

$\mathbb{Q}^{alg}[[a,b]] $ is not elementary equivalent to $\mathbb{C}[[a,b]]$, and the same for $\mathbb{Q}^{alg}[a]$ and $\mathbb{C}[a]$?

Since ACF is complete, $\mathbb{Q}^{\text{alg}}$ is elementary equivalent to $\mathbb{C}$, and by Ax-Kochen $\mathbb{Q}^{\text{alg}}[[a]]$ is elementary equivalent to $\mathbb{C}[[a]]$. But how should ...
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2answers
14 views

Let $R$ be a ring with $1$. If there exist disjoint comaximal ideals $I, J$, then for any $R$-module $M$, $M = IM \oplus JM$.

This questions is from a past year paper. Let $R$ be a ring with $1$. Suppose there exist distinct ideals $I, J$ of $R$ such that $R = I + J$ and $I \cap J = \{ 0 \}$. (a) Show that there exists $a \...
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0answers
18 views

A set which is closed under addition and is well-ordered

If I have a set of objects which support + and <, is that necessarily isomorphic to a subset of the real numbers? I ask this question because I'm trying to figure out the most general type of a ...
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1answer
23 views

Show that if N, K are normal subgroups of a group G, and N contains K then we have: $ G / N \cong (G/K) / (N /K) $ [duplicate]

Show that if $N, K$ are normal subgroups of a group $G$, and $N$ contains $K$ then we have: $$ G / N \cong (G/K) / (N /K) $$ Intuitively it looks correct, would like to know how I can approach this.
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0answers
42 views

Finding the generator of a cyclic group

Q) Show that $x$ or $2x$ is a generator of the cyclic group $(\mathbb{Z}_3[x]/\langle f(x)\rangle)^*$ where $f(x)$ is a cubic irreducible polynomial over $\mathbb{Z}_3$. My attempt: Let $F= \mathbb{Z}...
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0answers
12 views

Does ''direct sum '' in the definition of graded ring mean internal direct sum?

I just read the definition of graded ring on wiki. a graded ring is a ring that is a direct sum of abelian groups $R_{i}$such that $ R_{i}R_{j}\subseteq R_{i+j}$. Does it mean internal direct sum here?...
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0answers
52 views

Multiplication Table for $\Bbb Z_2[x] /<x^3 +x^2+x+1>$

The factor ring can be rewritten as $ \{a_o +a_1x+a_2x^2 \mid a_0,a_1, a_2 \in \mathbb{Z}_2 \}$. We figured out that the values on the multiplication table will be $\{0,1,1+x,1+x^2,x+x^2,x^2,1+x+x^2 \}...
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0answers
37 views

describe $\langle x^2, x+2\rangle$ as a subgroup

well, I totally have no idea what's the "describe" mean? the question is "describe $\langle x^2, x+2\rangle$ as a subgroup and an ideal of $\mathbb Z_4[x]$. anyone can describe the describe for me?
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1answer
24 views

Proper Ideals with Norm Relatively Prime to Conductor

Let $K$ be an imaginary quadratic number field, and $\mathcal{O}_K$ the ring of integers. Let $\mathcal{O}$ be an order. Call the $\textit{conductor}$ $f = [\mathcal{O}_K:\mathcal{O}]$. Given some $\...
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4answers
58 views

Number Theory theorem regarding ring of integers in $\mathbb{Q}[\sqrt{D}]$

Here is the theorem that I need to prove For $K = \mathbb{Q}[\sqrt{D}]$ we have $$\begin{align}O_K = \begin{cases} \mathbb{Z}[\sqrt{D}] & D \equiv 2, 3 \mod 4\\ \mathbb{Z}\...
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2answers
111 views

Construct a nonabelian group of order 44

Let $G$ be a group s.t. $|G|=44=2^211$. Using Sylow's Theorems, I have deduced that there is a unique Sylow $11$-subgroup of $G$; we shall call it $R$. Let $P$ be a Sylow $2$-subgroup of $G$. Then we ...
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2answers
71 views

''Homomorphism'' of rings

Let $R$ and $R'$ rings, $\phi$ a homomorphism of $R$ to $R'$ such that $\phi(x+y)=\phi(x)+\phi(y), \quad \forall x,y \in R$. $\phi(xy)=\phi(x)\phi(y) \quad or \quad \phi(xy)=\phi(y)\phi(x) \quad \...
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1answer
31 views

Show that an infinite abelian group all of whose proper quotients are finite $\cong \mathbb{Z}$

Show that an Infinite abelian group all of whose proper quoteints are finite $\cong \mathbb{Z}$ So I'm a little confused on how to get started here. Perhaps something like: Let $x \in G$, then $|G/&...
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0answers
29 views

Let $G$ be a group s.t. $|G|=pqr$ where $p,q,r$ are primes that need not be distinct. Prove that $G$ is soluble.

Let $G$ be a group s.t. $|G|=pqr$ where $p,q,r$ are primes that need not be distinct. Prove that $G$ is soluble. So, I don't know whether I should handle this case by case and try to get the Sylow ...
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2answers
34 views

Corresponding space to a irreducible part of a given representation of $S_n$

My setting is the following: I have an action of $S_n$ on some $\mathbb{C}$-vector space, $V$, by permuting a special basis: $$<a^1_{1},\dots,a^1_{n!},a^2_{1},\dots,a^2_{n!},\dots,a^k_{n!},b_1,\...
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0answers
15 views

Galois group as a subgroup of $S_4$ relation to constructible algebraic element

I have that $\gamma$ is a constructible element of degree $4$ over $\mathbb{Q}$, ie. $[\mathbb{Q}(\gamma):\mathbb{Q}] = 4$. Let $N$ be the normal closure of $\mathbb{Q}(\gamma)$ over $\mathbb{Q}$. The ...
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1answer
27 views

A linear transform mapping one billinear map to another may not exist (counter-example)

$$\newcommand{\im}{\mathrm{im}\;}\newcommand{\Span}{\mathrm{Span}}\newcommand{\rank}{\mathrm{rank}\;}$$For a bilinear map $\phi : V \times W \to E$ define the first nullspace as $$ N_1(\phi) = \{ v \...
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2answers
41 views

Quotient group of dihedral group

Let $G=\{e,r^{2},...,r^{8},s,sr,...,sr^{8}\}$ and let $N=\langle r^{3} \rangle.$ Now let $\pi(g)=\bar{g}=gN$ be surjective with kernel $N$. I have to show that $G/N=\{\bar{e},\bar{r},\bar{r^{2}},\...
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1answer
39 views

Is there a general and mechanical method to solve algebra of sets (or alg.of propositions) equations?

In some simple cases it seems possible to solve for X a set equation. For example, if I am given : X Inter U = U , and knowing the law according to which S Inter U = S for any set S, I can find ...
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0answers
10 views

skew-symmetric non-degenerate bilinear space has even dimension [duplicate]

How to prove skew-symmetric non-degenerate bilinear space has even dimension with skew-symmetric defined as $(x,y) = - (y,x), \forall x,y \in V$, where $V$ is the vector space
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0answers
31 views

Galois Extensions

I was trying to answer the following question and wasn't sure how to proceed. Which one of the following extensions $K \subset L$ is not Galois? (a) $K = \mathbb{Z}_3(x)$ and $L = K[a]/(a^3-x)$ ...
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1answer
22 views

Can anyone gives me an example of $V$ such that there exists at least one pair of elements $ a,c(\neq 0 ,1) \in D$ such that $V(a) > V(ac)$

$D$ is an Integral Domain. $V$ only satisfies the first Euclidean property, i.e. for all $\,a,b\in D\,$ if $\,b\neq 0\,$ then there are $\,q,r\in D\,$ such that $\, a = qb +r\,$ with either $r=0$ or $...
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1answer
33 views

If $R$ is a right-Noetherian ring and the Jacobson radical satisfies the right Artin-Rees property, then $\bigcap^{\infty}_{n=1}\text{Jac}(R)^n = 0$

A (two-sided) ideal $I$ of a ring with identity $R$ has the right Artin-Rees property if for any right-ideal $E$ of $R$, there exists an integer $n\geq1$ such that $E\cap I^n\subseteq EI$. If $R$ ...
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1answer
41 views

Prove that if $p$ is a prime number and $n$ is a positive integer, then $\phi(p^n ) = p^n − p^{n−1 }$. [on hold]

Would I have to use Euler's phi function or the Euler Fermat theorem? Any help appreciated!
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0answers
14 views

What is the intuition behind $C^*$-algebras?

A Banach $*$-algebra $A$ is a Banach space such that it is also an algebra and the product is submultiplicative w.r.t. the norm and it has an involution, i.e., an autiautomorphism of order $2$. If in ...
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1answer
29 views

Finding smith normal form of $x-A$

$$ A = \quad \begin{pmatrix} 1 & 1 &0 &0 \\ -1 & -1 & 0& 0\\ -2&-2 & 2 & 1 \\ 1& 1&-1 & 0 \end{pmatrix} \quad $$ I want to find Smith normal form of ...
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0answers
36 views

Why does group action by conjugation on Sylow subgroups define a representation

I am studying representations and I am stumbled upon this: Take the dicyclic group of order $12: G=\Bbb Z/3\Bbb Z \rtimes \Bbb Z/4\Bbb Z$. The action of G on its 2-Sylow subgroups appearantly gives ...
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1answer
23 views

Find $ker (\phi)$ and $Im(\phi)$ and create the operation $ Q/ ker(\phi)$?

Let us consider the Quartenion group $Q=\{ \pm 1, \pm i, \pm j, \pm k \}$ and $N=\left\langle j \right\rangle$ in $Q$ be a cyclic subgroup. Define $ \phi: Q \to \mathbb{Z}_2 \oplus \mathbb{Z}_2 $ by $ ...
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1answer
32 views

Can anyone give me an example of an Euclidean Domain $D$ whose Euclidean Valuation at a point $a$: $v(a)$ is bigger than $\min \{v(ax) : x\in D\}$?

Can anyone give me an example of an Euclidean Domain $D$ whose Euclidean Valuation at a point $a$ , $v(a)$ is bigger than $\min \{v(ax) : x\in D\}$ ? I know that second condition for being Euclidean ...
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0answers
33 views

Eigenvalue of matrix over (Z/pZ)

How can I show for $M\in\text{GL}_d(\mathbb{Z}/p\mathbb{Z})$ with $\text{ord}(M)=p^n$ ($n$ a positive integer), that $1$ is an eigenvalue of $M$? I would be grateful for any hint or advice. Thank ...
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2answers
46 views

A proof about Automorphism in congruence class

Suppose $gcd(m,n)=1$, and let $F :Z_n→Z_n$ be defined by $F([a])=m[a]$. Prove that $F$ is an automorphism of the additive group $Z_n$. I find it is diffcult to prove $F$ is injective and surjective. ...
3
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1answer
186 views

Is there a canonical “inverse” of Abelianization?

We know that the abelianization of a free product is the direct sum, for example $Ab(\mathbb{Z}*\mathbb{Z})=\mathbb{Z}\oplus\mathbb{Z}$. Is there a “canonical” (or even non-canonical) operator that ...
2
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0answers
26 views

Question on irreducible character of regular representation of the symmetric group

Consider the symmetric group $S_n$ acting on $A=\{1,..,n\}$, for any nonnegative integer $k\leq n/2$, denote $A_k$ to be the collection of all $k$-element subsets of $A$. Let $\chi_k$ be the character ...
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0answers
19 views

Using Zassenhaus to obtain an isomorphism of bigraded groups

Let $G$ be a group. Suppose that $F^{\bullet}G$ is a filtration on $G$. If $q: G \to K$ is a quotient map, then we get an induced filtration of $K$ given by $F^aK:= q(F^aG) = (F^aG)/(F^aG \cap \ker q)$...
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3answers
55 views

Suppose that $H_1/H_2$ is Abelian. Show that $H_1 N / H_2 N$ is Abelian.

Suppose $G$ is a group and $H_1$, $H_2$, $N$ are subgroups of $G$. $N$ is a normal subgroup of $G$ and $H_2$ is a normal subgroup of $H_1$. Suppose that $H_1/H_2$ is Abelian. Show that $H_1N/H_2N$ is ...
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2answers
23 views

Is the restriction to $H$ applied on both $G, N$ or only on $G$?

I'm reading Ash's Basic Abstract Algebra. Here: I got a - perhaps - useless question: When we restrict $G\to G/N$ to $H$, do we consider this restriction on both $G,N$ or only on $G$? Perhaps it ...