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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Minimum possible size of generating set of $(\mathbb{Z}_p)^m$

Is it true that the group $(\mathbb{Z}_p)^m$ cannot be generated by less than $m$ elements?
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2answers
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Even permutation in Sn

How to show that if $\pi \in S_n$ is a square then $\pi$ is an even permutation. Is the converse statement true: each even permutation is a square?
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7 views

$K_1(R)$ and splitting

Let $R$ be a commutative ring with unit. Consider the exact sequence $1\to E(R)\to Gl(R)\to K_1(R)\to 1$. Under what conditions does this exact sequence split?
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19 views

Square element in a cyclic group

Which elements of a cyclic group are squares (an element $g$ of a group $G$ is a square if $g=h^2$ for some $h \in G$)? Solution: Let $G = \{ 1,a,a^2, \ldots , a^n \}$ If $n$ is even, then $1,a^2, ...
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Motivation For Tensor Product of R-Modules

I have recently learned about tensor products of modules,specifically the material in Dummit and Foote chapter 10 section 4. My understanding is that the construction of tensor spaces is important ...
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1answer
33 views

Squares in a field with $q^n$ elements.

Let $F$ be a field with $q^n$ elements, where $q$ is an odd prime. Write $q^n=2m +1$ with $m \in \mathbb{N}.$ If $r \in F^{\times},$ show that the equation $y^2= r$ has a solution iff $r^m=1.$
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Riddle: Assigning Students into Groups

Suppose you had a classroom with 25 students. You want to assign 6 homework assignments over the course of the term and for each of these assignments students will work in groups of 5. But you want to ...
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1answer
20 views

Provide an example to show that $S$ may not necessarily be a unique factorisation domain when $R$ is a unique factorisation domain.

Let $R$ and $S$ be integral domains, and suppose that $\phi:R \rightarrow S$ is a surjective ring homomorphism. Provide an example to show that $S$ may not necessarily be a unique factorisation domain ...
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31 views

Show that $Im(\phi) = \mathbb{Z}[i]$

Let $\phi: \mathbb{Z}[x]\to \mathbb{C}$ and $\phi(f(x)) = f(i), \forall f(x) \in \mathbb{Z}[x].$ Show that $Im(\phi) = \mathbb{Z}[i]$ My attempt: I am not sure if it's correct: First, we need to ...
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8 views

Show that the positive units of $K$ form a group isomorphic to $\mathbb Z$

Let $K$ be a cubic field such that $r_1=r_2=1$. Suppose $K$ is imbedded in $ \mathbb R$ ($r_1$ and $r_2$ are the usual notations, $r_1$ denotes the number of isomorphisms $\sigma : K \to \mathbb R$ ...
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How do I prove that $\forall \beta\in F(\alpha)\setminus F$ is transcendental?

Let $E/F$ be a field extension. Let $\alpha\in E$ be transcendental over $F$. Let $\beta\in F(\alpha)\setminus F$. Then, how do I prove that $\beta$ is transcendental over $F$? Here's how I tried: ...
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3answers
51 views

How many elements does the free product $\mathbb{Z}/2\mathbb{Z}\ast\mathbb{Z}/2\mathbb{Z}$ have?

Taken from Hatcher, I think the free product $G =\mathbb{Z}/2\mathbb{Z}\ast\mathbb{Z}/2\mathbb{Z}$ should have infinite elements taking the form of words containing alternate elements, i.e. $a, b, ab, ...
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30 views

“Breaking the symmetry” in solving algebraic equations

I've heard somewhere a discussion about solving algebraic equations before: When solving a quadratic equation, we are essentially doing the following. Observe that ...
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35 views

How many elements are in $\mathbb{Z}_5[i]/\langle1+i\rangle$

Maybe this question has been asked by someone else before, but I could not find a duplicate and I would really appreciate some help. I know that an element of $\mathbb{Z}_5[i]/\langle1+i\rangle$ is ...
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3answers
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If α is algebraic over K then all the elements of K(α) are algebraic over K

Let $L$ : $K$ be a field extension, $\alpha \in L$ algebraic over $K$. Show that every element of $K(\alpha)$ is algebraic over $K$, where $K(\alpha)$ is the smallest subfield that contains $\alpha$ ...
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1answer
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Free action on space implies that each point has a neighborhood that has an empty intersection with translations

Suppose $G$ is a topological group, $X$ a topological space and $G \times X \rightarrow X$ group action that is continuous. Further, suppose that the action is free ($G_x = \{e\}$, for all $x$). What ...
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1answer
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$a$ and $1+a^{-1}$ have same degree over $F$ if $a$ is algebraic over $F$

Suppose that $a$ is algebraic over a field $F$. Show that $a$ and $1+a^{-1}$ have the same degree over $F$. Not really sure where to start for this one. I know that I have to show that ...
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2answers
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Cyclic group Zp [duplicate]

How to show if $p$ is prime, then group $Z_{p}^{*}$ is cyclic. Tip Let $g$ and $h$ of a commutative group $G$ have orders $n$ and $m$ respectively. There exists and element $x \in G$ of order $LCM ...
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Let $a$ be a complex zero of $x^2+x+1$ over $\mathbb{Q}$. Prove that $\mathbb{Q}(\sqrt{a})=\mathbb{Q}(a)$.

Let $a$ be a complex zero of $x^2+x+1$ over $\mathbb{Q}$. Prove that $\mathbb{Q}(\sqrt{a})=\mathbb{Q}(a)$. Let $f(x)=x^2+x+1$. Then $f(a)=a^2+a+1=0$. To show equality of the two fields, we need ...
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0answers
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Jacobson radical of polynomial ring [duplicate]

Let $R$ be a ring, i want to show that: if R has not nil-ideals than $J(R[x]) = \left\{ \emptyset \right\}, \text{where $J$ Jacobson radical, $R[x]$ - polynomial ring over $R$}$
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1answer
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Prove that if $R$ is a principal ideal domain, then either $R \cong S$, or $S$ is a field.

Let $R$ and $S$ be integral domain and suppose that $\phi: R \rightarrow S$ is a surjective ring homomorphism. Prove that if $R$ is a principal ideal domain, then either $R \cong S$, or $S$ is a ...
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3answers
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Show that $N = f(M) \bigoplus N'$ if and only if there exists $\alpha: N \rightarrow M$ such that $\alpha(f(m)) = m$ for all $m \in M$.

Let $R$ be a ring with $1$, and let $f: M \rightarrow N$ be an $R$-module homomorphism. Suppose that $f$ is injective. Show that $N$ has a submodule $N'$ such that $N = f(M) \bigoplus N'$ if and only ...
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If $|G|=p^n$, then $p^2 \le |G : G^\prime|$.

Prove that, if $G$ be a p-group of order $p^n$, then $p^2 \le |G : G^\prime|$, where $G^\prime$ is the commutator subgroup of $G$ and $n \ge 2$.
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Artin-Rees Lemma for Semigroups [on hold]

(Artin-Rees Lemma) Let $S$ be a Noetherian semigroup and $A,B$ be ideals of $S$. A∩B^i=(A∩B^N)B^(i-N) for each İ≥N where N is a natural number. Does anyone know its proof? Hi, I use İ and N as a ...
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Show that there exists a $3 × 3$ invertible matrix $M$ with entries in $\mathbb{Z}/2\mathbb{Z}$ such that $M^7 = I_3$.

Show that there exists a $3 × 3$ invertible matrix $M$ (which is not the identity matrix) with entries in the field $\mathbb{Z}/2\mathbb{Z}$ such that $M^7 = $Identity matrix. All I could do was use ...
3
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1answer
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Let $a$, $b$, and $c$ be elements of a commutative ring, and suppose $a$ is a unit. Prove that $b$ divides $c$ if and only if $ab$ divides $c$.

Let $a$, $b$, and $c$ be elements of a commutative ring $R$, and suppose $a$ is a unit. Prove that $b$ divides $c$ if and only if $ab$ divides $c$. Okay so here is the proof I came up with. Please be ...
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Generating sets of $Q_8$

Consider the matrices $$I=\left( {\begin{array}{ccc} 1 & 0 \\ 0 & 1 \\ \end{array} } \right), \qquad A=\left( {\begin{array}{ccc} i & 0 \\ 0 & i \\ \end{array} } ...
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Showing that $40^{\circ}$ is not constructible

Show that $40^{\circ}$ is not constructible. Attempt We note that $\cos 120^{\circ}=-\frac{1}{2}$ and that it also equals $4\cos^340^{\circ}-3\cos40^{\circ}$, which is obtained by using the ...
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2answers
38 views

If the localizations of two submodules with respect to any prime ideal are equal then the submodules are equal [on hold]

I want to prove the following: Let R be a commutative ring with 1 and let N and L be two submodules of an R-module M. If the localizations of N and L with respect to any prime ideal of R are ...
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0answers
26 views

Subgroups of finite index have finitely many conjugacy classes

is it true the following statement: Let $G$ be a group and let $H$ be a subgroup of $G$. If the index $[G:H]$ of $H$ in $G$ is finite, then $H$ have finitely many conjugacy classes. What I think, is ...
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2answers
42 views

Let $R$ be a finite ring with unity. Prove that $x$ is a LZD $\iff$ x is a RZD

Let $R$ be a finite ring with unity. Let $x \in R$. Prove that $x$ is a Left Zero Divisor $\iff$ x is a Right Zero Divisor. My attempt Suppose $x$ is a LZD. Then, $\exists y \in R$ such that $xy = ...
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1answer
53 views

Ring Homomorphisms from $\mathbb Z_{20} \to \mathbb Z_{30}$

We need to find all ring homomorphisms from $\mathbb Z_{20} \to \mathbb Z_{30} $ ; I read its solution somewhere which states that : $R : \mathbb Z_{20} \to \mathbb Z_{30}$ defined by $R(x) = ax$ , ...
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0answers
14 views

An R-matrix in a quasitriangular Hopf algebra

I am new to the theory of Hopf algebra. So I am sorry if the following question has really trivial answer. Suppose that we have a quasi triangular Hopf algebra with the universal R-matrix $R$. It ...
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38 views

A ring with a left cancellable element and a right identity always has an identity.

Let $R$ be a ring with $a, e \in R$ such that $a$ is not a left zero-divisor and $be=b, \forall b \in R$. Prove that $R$ has an identity. My attempt Let, $aeb = ab \Rightarrow aeb - ab = 0 ...
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What would be a value of $X$ under an automorphism of $F(X)$ over $F$?

Let $\sigma:F(X)\rightarrow F(X)$ be a field automorphism fixing $F$. What would be an value of $\sigma(X)$? Since $X$ is transcendental over $F$, $\sigma(X)$ is a transcendental over $F$ and ...
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0answers
36 views

Is the ring of germs of $C^\infty$ functions at $0$ Noetherian?

I'm considering the property of the ring $R:=C^\infty(\mathbb R)/I$, where $I$ is the ideal of all smooth functions that vanish at a neighborhood of $0$. I find that $R$ is a local ring of which the ...
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1answer
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To prove that $I = \{\,(n,m) \in \Bbb Z \times \Bbb Z : n,m $ are even $\}$ is not a maximal ideal of $ \Bbb Z \times \Bbb Z $.

To prove that $I = \{\,(n,m) \in \Bbb Z \times \Bbb Z : n,m $ are even $\}$ is not a maximal ideal of $ \Bbb Z \times \Bbb Z $. Thus there exists an ideal $J$ of $ \Bbb Z \times \Bbb Z $ such that $I ...
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how many proper subgroups are there in a trivial group{e}? [on hold]

Is there any proper subgroup of a trivial group of order 1? If yes or no how to prove it?
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1answer
31 views

Ring Homomorphism from $\mathbb{Z}_m$ to $\mathbb{Z}_n$

Suppose $R$ is a ring homomorphism from $\Bbb{Z}_m$ to $\Bbb{Z}_n$ , prove that if $R(1) = a$ then $(a^2)=a$. Also show, its converse is not true. The first part goes like this : $R(1) = a , ...
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1answer
34 views

Proving every element of $\mathbb{F}[x]/(p(x))$ can be expressed uniquely

Proving every element of $\mathbb{F}[x]/(p(x))$ can be expressed uniquely in the form $a(x) + (p(x))$ where $\text{deg}(a) < \text{deg}(p)$ this is a homework problem and I'm stuck, here is my ...
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1answer
58 views

When is $HK \cong H \times K$?

Suppose $G$ is a group and $H$ and $K$ are subgroups such that $G = HK$ and $H \cap K = \left\{e\right\}$, the identity element of $G$. When can we say that $HK \cong H\times K$? I tried to set up ...
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1answer
11 views

Are these two definitions of separable extension equivalent?

Definition1 - wikipedia Let $E/F$ be an algebraic extension. Then $E/F$ is separable iff for each $\alpha\in E$, the minimal polynomial of $\alpha$ over $F$ is separable. Definition 2 - ...
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Let $I = \{a +\sqrt2b \in \Bbb Z[\sqrt2] : a$ and $b$ are both multiple of $5\}$. [duplicate]

Let $I = \{a +\sqrt2b \in \Bbb Z[\sqrt2] : a$ and $b$ are both multiple of $5\}$. Show that $I$ is a ideal.(I have done it) But how to show that it is maximal?
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1answer
13 views

Commutative rings and ideals, showing a map is well defined

Let $R$ be a commutative ring with an ideal $I$. The additive group $R/I$ is the set of cosets of $I$ with respect to addition in $R$. Let $\cdot : R/I \times R/I \to R/I$ be defined by ...
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What is the difference between the middle factor and the middle term of permutation ? [duplicate]

What is the difference between the middle factor and the middle term of permutation ?
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1answer
25 views

Restriction of topological ring isomorphism

If $\theta: R\to S$ is an isomorphism of topological rings then do we obtain a topological group isomorphism $\theta|_{R^{\times}}:R^{\times}\to S^{\times}$ by restricting to their groups of units? ...
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scalar multiplication in koblitz curve cryptography [on hold]

please let me know how to find the reduced tau-adic form of given scalar. i.e $$K=K(\mod \tau^n-1)= (K_{n-1} ,K_{n-2},\dots, K_1,K_0)\tau$$ ?? where τ=tau
2
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1answer
18 views

Proving that a Left semisimple ring $R$ is both left noetherian and left artinian

Prove that a left semi-simple ring $R$ is both left noetherian and left-artinian. I am following the proof given in pg 27,A first course in non-commutative rings (T.Y.Lam). Its strategy is to show ...
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1answer
45 views

Let $I = \{a +ib \in \Bbb Z[i] : a$ and $b$ are both multiple of $5\}$.

Let $I = \{a +ib \in \Bbb Z[i] : a$ and $b$ are both multiple of $5\}$. Show that $I$ is an ideal of $\Bbb Z[i]$. Is $I$ a maximal ideal? And to find the numbers of elements of the quotient ring $\Bbb ...
2
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2answers
43 views

Is this group homomorphism well-defined?

Let $X = \langle a, b \mid aba^{-1}b^{-1} \rangle$ and $Y = \langle a, b \mid aba^{-1}b \rangle$. I want to define $f : X \rightarrow Y$, such that, $f(a) = a$ and $f(b) = b^2$, however I'm having ...