Abstract algebra is the study of algebraic objects. Some of the more common algebraic objects are groups, rings, fields, vector spaces, modules, and other advanced topics.

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Decomposition of a cycle as a product of transpositions

Can someone please explain the rules pertaining to different ways to write a cycle decomposition as products of 2-cycles, an example from textbook: I understand this $$ (12345) = (54)(53)(52)(51) ...
-2
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1answer
151 views

Endomorphisms of Noetherian and Artinian modules [duplicate]

Let $A$ be a commutative ring with unit and $M$ be an $A$-module. Let $f: M\longrightarrow M$ be a module homomorphism. Then (1) if $M$ is Noetherian and $f$ surjective, then $f$ is bijective. (2) ...
2
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3answers
107 views

Prove $\forall r \in \mathbb{R}. \exists k \in \mathbb{Z}. r < k$

I would like to prove that for every real number there exists an integer that is greater than it. My problem lies in that I am not sure how to construct the real numbers and provide their theory with ...
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1answer
51 views

How to find a subspace with maximum dimension that doesn't include $k$ special vectors?

Assume $V_n = \{(a_1,a_2,\cdots,a_n), a_i \in \mathbb{GF}(2)\}$ and $k$ linearly independent vectors of $V_n$. How to find a subspace with maximum dimension that doesn't include these $k$ vectors?
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2answers
75 views

Is it necessarily true that if a polynomial is irreducible in $\mathbb Z_n$ ($n$ is prime) then it is irreducible in $\mathbb{Q}$?

I have played around with a couple examples and I've consistently seen a pattern where the polynomials that are irreducible in $\mathbb{Z}_n$ are irreducible in $\mathbb{Q}$. Can anyone challenge this ...
3
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1answer
227 views

List all cosets of H and K

Let $G = \mathbb Z_3 \times \mathbb Z_6$, $H = \langle (1,2)\rangle$ and let $K = \langle (1,3)\rangle$. List all cosets of $H$ and $K$. Can somebody please explain me how to do this problem. ...
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2answers
128 views

group generators of $(\mathbb Z_{17}-\{0\},\times)$

How to find generators of $(\mathbb Z_{17}-\{0\},\times)$? Is there a faster way to find generators than trying every element in the group? I know that for additive group, if a number say m is ...
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0answers
120 views

Another galois theory problem. (quartic with dihedral Galois group)

Given: If $f$ in $k[X]$ is an irreducible quartic and $G$ is the galois group of its splitting field, then $G$ is contained in $D_8$ iff the resolvent cubic of $f$ has a root in $k$. Now suppose ...
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1answer
82 views

Symmetries- Table of group of symmetries

How do you go about listing the symmetries of say letters V or Z or along that line and their tables? Its too vague for me to understand what they are asking for i know the given theorem requires 4 ...
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4answers
603 views

If a subring of a ring R has identity, Does R also have the identity?

I know it does not make sense that if a subring of a ring R is commutative, then R is also commutative. (For example, the set consisting of the matrices whose all entries except (1,1)-entry are zero, ...
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0answers
63 views

Order of elements of a group.

Assume that G is an abelian group, and a∈G. (a) Assume that |a|=r and that m|r, say r=mt. Prove that $|a^t|$=m. Proof:Assume a∈G and |a|=r and m|r, say r=mt. Assume $|a^t|$=k Since ...
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2answers
128 views

Confused by Example in Herstein's “Topics in Algebra”

The following comes from I.N. Herstein's "Topics in Algebra", just after defining subgroups. He gives the following example Let $S$ be any set and $A(S)$ be the set of one-to-one mappings of $S$ ...
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1answer
205 views

Prime Number Theorem in $\mathbb{F}_p[x]$

What is the probability that a randomly chosen monic polynomial of large degree $n$ in $\mathbb{F}_p[x]$ is irreducible? We can interpret this probability as ...
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4answers
257 views

Infinite coproduct of abelian groups

One can see on every text (book, lesson, comments) that a direct sum/coproduct of abelian groups is the same as a finite product but in the infinite case, the direct sum/coproduct is only a subgroup ...
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2answers
68 views

I don't think I'm using an assumption in this proof. Anything wrong?

Define the exponent $\exp(G)$ of a finite group $G$ to be the smallest positive integer $k$ such that $g^k = e$ for all $g \in G$. The question asks If $G$ is a finite abelian group, prove that ...
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0answers
42 views

Order of $5+2^k\mathbf{Z}$ in $(\mathbf{Z}/2^k\mathbf{Z})^*$

When I test for small values of k, it seems that the order of $5+2^k\mathbf{Z}$ in $(\mathbf{Z}/2^k\mathbf{Z})^*$ is $2^{k-2}$. How do I prove this?
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1answer
82 views

There is no “operad of fields”

I've read the following proof-less claim: there is no operad such that the algebras over it are fields. We can make that precise by asking whether there's an operad $\mathcal{P}$ in abelian groups ...
7
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1answer
128 views

A valuation-like function $w: \mathbb{N}^{+} \rightarrow \mathbb{N}$ is a $p$-adic valuation?

This question is a variant of problem 4, pg. 21, from Birkhoff and Maclane, A Survey of Modern Algebra. Given a function $w: \mathbb{N}^+ \rightarrow \mathbb{N}$ that behaves like a valuation ...
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1answer
27 views

Proof: If $r \in R$ is irreducible then $ur$ is irreducible where $u$ is a unit.

If $r \in R$ is irreducible then $r=ab, a,b \in R$ implies $a$ or $b$ is a unit. How does one proof $ur$ is irreducible if $u$ is a unit. I must proof: $ur = mn, m, n\in R$ then $m$ or $n$ is a ...
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2answers
849 views

What are the differences between Jacobson's “Basic Algebra” and “Lectures in Abstract Algebra”?

Nathan Jacobson's books "Basic Algebra I, II" and "Lectures in Abstract Algebra - Volumes I, II, III (GTM 30, 31, 32)". What are the differences between these two books? 1) The subject. The ...
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1answer
78 views

What can conclude about $[G:H]$?

Assume that $H$ is a subgroup of a finite group $G$, and that $G$ contains elements $a_1, a_2,...,a_n$ such that $a_i a_j^{-1} \notin H $ for $1\leq i < n, 1 \leq j <n $, and $i \neq j$. What ...
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1answer
95 views

Which of the following statement is not necessarily true for the product of rings $R \times R$ when it is true for $R$?

$R$ is a ring. Which of the following statements is not necessarily true for the product of rings $R \times R$ when it is true for $R$? A. There exists some generator whose order is finite. B. $R$ ...
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1answer
64 views

Find the number of cosets$ [G:H] $?

Assume that $G$ is a cyclic group of order $n$, that $G =\ <a> $, that $k|n$ , and that $H=<a^k>$. Find $[G:H] $ the number of cosets to the subgroup H I think that since $k|n$ ...
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2answers
315 views

Normal Ring and Prime Ideal whose Square is Principal

Let $k$ be a field of characteristic $\neq2$ and consider $R=k[x,y]/(y^2-x^3+x)$. Then (a) Show that $R$ is normal. (b) Let $P=(x,y)$ be a prime ideal of $R$. Show that $P^2$ is a principal ideal. ...
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3answers
377 views

Groups : $a$ and $b$ commute, prove $a^2$ commutes with $b^2$

If $a$ and $b$ are in $G$ and $ab=ba$ we say that $a$ and $b$ commute. Assuming that $a$ and $b$ commute prove the following: 1) $a^2$ commutes with $b^2$
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2answers
76 views

Prooving by absurd that $d \nmid 4,5,10, 20$

What I'm trying to solve is the following: Given that $(a:b) = 2$, proove that $(a^2 + 2b^2+10:20) = 2$. So, basically, I think that what I need to do is to show that if $d = a^2 + 2b^2 + 10$, then ...
2
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1answer
57 views

Using roots of irreducible polynomials to rewrite products.

Suppose $F$ is a field, and $p(x)\in F[x]$ is irreducible, of degree $n$, with a root $\alpha$. "$F(\alpha)$ is closed under multiplication since $\alpha^n,\alpha^{n+1},\ldots $ can be written as ...
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1answer
114 views

Euler function and $\mathbb{Z}/n\mathbb{Z}$

I am trying to solve a very interesting problem about the ring $\mathbb{Z}/n\mathbb{Z}$ and Euler function $\phi (n)$, but i am not sure how to start, i have a few ideas, but none of them leads me to ...
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5answers
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The ring $ℤ/nℤ$ is a field if and only if $n$ is prime

Let $n \in ℕ$. Show that the ring $ℤ/nℤ$ is a field if and only if $n$ is prime. Let $n$ prime. I need to show that if $\bar{a} \neq 0$ then $∃\bar b: \bar{a} \cdot \bar{b} = \bar{1}$. Any ...
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1answer
51 views

Proving if $H<G$ and $N\triangleleft$ G then $H\cap N \triangleleft H$

I want to prove that if $H<G$ and $N\triangleleft$ G then $H\cap N \triangleleft H$. Here is what I have done: $N\triangleleft G \iff aN=Na \quad \forall a\in G$ $hN=Nh \quad \forall h\in H$ ...
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1answer
77 views

Can one prove $(g_1H)(g_2H) = (g_1g_2)H$, $g_1, g_2 \in G$ if and only if $gH = Hg$ for all $g \in G$?

Let $G$ be a group and $H$ a subgroup of $G$. I have proven in an exercise that $gH = Hg$ for all $g \in G$ implies $(g_1H)(g_2H) = (g_1g_2)H$, $g_1, g_2 \in G$ where $(g_1H)(g_2H) = (xy | x\in g_1H, ...
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1answer
251 views

Decomposition into direct sum of fields? [True or false]

I am stuck in a 'true or false' question about decomposition into direct sum of finite fields and don't really know how to prove the problem. Can anybody give me a hint or an idea how to solve it, ...
2
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0answers
176 views

Wedderburn decomposition of $D_{5}$

I'm wondering how to solve this question. Find the Wedderburn decomposition of $D_{5},$ the dihedral group of order 10, over the field $\mathbb{F}_{3}.$ I have shown that the irreducible ...
3
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2answers
164 views

How many elements are there in the intersection of two subgroups of a finite cyclic group?

Let's assume that we have two subgroups $H_1$ and $H_2$ in $\mathbb{Z}_n$ with $k$ and $l$ elements respectively. How many elements are there in the intersection $H_1\cap H_2$? Let denote this by ...
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0answers
119 views

Connected subgroups of SU(2) and SU(3)

I am reading 'Lie groups, Lie Algebras, and Representations : An Introduction' by Brian Hall and am unable to do the problem 17 in chapter 3. It says Show that every connected Lie subgroup of ...
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3answers
90 views

divisibility question in abstract algebra over a field

$d|n \Rightarrow x^{p^d} - x$ divides $x^{p^n} - x$ over $\mathbb{F}_{p}$ where $\mathbb{F}$ is a field. Attempt: $d|n \Rightarrow x^{p^d} - x$ divides $x^{p^n} - x \Rightarrow x^{p^d - 1} - 1$ ...
2
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4answers
45 views

Ideals in $Z_{24}$

The ideals in $Z_{24}$ are $(\overline{0}), (\overline{12}), (\overline{8}), (\overline{6}), (\overline{4}), (\overline{3}), (\overline{2})$ and $Z_{24}$ itself. Now why isn't, say, ...
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2answers
533 views

Is it possible to have a non-trivial homomorphism of some finite group into some infinite group?

Let $G$ be a group of some finite order, and let $G^\prime$ be some group of infinite order. Then there is the trivial homomorphism of $G$ into $G^\prime$ which maps each element of $G$ into the ...
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2answers
115 views

Proving that a discrete valuation-like function $w: \mathbb{Z}\backslash\{0\} \rightarrow \mathbb{N}$ is a $p$-adic valuation

This problem is from Birkhoff and Maclane, A Survey of Modern Algebra, pg 21, problem 4*. Given a function $w: \mathbb{Z}\backslash\{0\} \rightarrow \mathbb{N}$ that behaves like a discrete ...
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0answers
39 views

Idempotents in a ring of fractions of the tensor product of Gaussian integers [duplicate]

Let $S=\{x^0,x^1,x^2,...\}\subset \mathbb{Z}$ be the multiplicatively closed subset generated by $x$. What are the nontrivial idempotents in the total quotient ring ...
3
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2answers
135 views

Writing real invertible matrices as exponential of real matrices

Every invertible square matrix with complex entries can be written as the exponential of a complex matrix. I wish to ask if it is true that Every invertible real matrix with positive determinant ...
2
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1answer
173 views

When are $\mathbb Z_m$ and $\mathbb Z_n$ homomorphic?

Let $m$ and $n$ be two given positive integers such that $m<n$. Then what are the necessary and sufficient conditions for the groups $(\mathbb Z_m,+_m)$ and $(\mathbb Z_n,+_n)$ to be homomorphic ...
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3answers
483 views

If a group $G$ has odd order, then the square function is injective.

Suppose $G$ has odd order, show the function $f:G\rightarrow G$ defined by $f(x)=x^2$ is injective. This proposition is easily provable if we assume $G$ is Abelian, but I don't know how to start this ...
2
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2answers
52 views

In the field $\mathbb{Z}_7[x]/\langle x^4+x+1\rangle$, find the inverse of $f(x)=x^3+x+3$.

In the field $\mathbb{Z}_7[x]/\langle x^4+x+1\rangle$, find the inverse of $f(x)=x^3+x+3$. I know how to find the inverses of elements within sets, rings, and fields. I know what to do if the field ...
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1answer
104 views

$GL_n(F)$ is not abelian for $n\ge 2$ counter example?

So I am asked to prove that for $n \ge 2$, the group $GL_n(F)$, where $F$ is any field, is non-abelian. I figure this amounts to finding a counter-example for all such $n$. It wasn't hard, but I'm ...
2
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1answer
98 views

Quantum Hamiltonian commuting with the Pauli-Runge vector.

I have to prove that $[A_j, H] = 0$, with; $$\vec{A} = \frac{1}{2Ze^{2}m}(\vec{L} \times \vec{P} - \vec{p} \times \vec{L}) + \frac{\vec{r}}{r}$$ $$H = \frac{p^2}{2m} - \frac{Ze^2}{r}$$ And, $Z, e, ...
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2answers
101 views

The polynomial $p(x)=x^4+x+1$ can be shown to be irreducible over $\mathbb{Z}_7$. Show that $\mathbb{Z}_7[x]/\langle p(x)\rangle$ is a field.

The polynomial $p(x)=x^4+x+1$ can be shown to be irreducible over $\mathbb{Z}_7$. Show that $\mathbb{Z}_7[x]/\langle p(x)\rangle$ is a field. Since $p(x)$ is irreducible over $\mathbb{Z}_7$, then ...
0
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1answer
168 views

Is a congruence (equivalence) class modulo n a group?

I know that the set of all equivalence classes Z/nZ is a group (with identity element the equivalence class [0], inverse element -[a]=[n-a]=[-a], etc.). However, is a single equivalence class modulo ...
2
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3answers
332 views

Boolean rings have characteristic $2$

Let $R$ be a ring such that $a^2=a$ for all $a\in R$. Show that $a+a=0$ for all $a\in R$. I don't really understand what to do here. The only way that this would be possible is if $a=0$. So $R$ ...
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2answers
96 views

Is $\mathbb{Z}$ a commutative ring?

Is $\mathbb{Z}$ a commutative ring? If so, would this imply that $\phi:\mathbb{Z}\to\mathbb{Z}$ is a commutative ring isomorphism? I know that $\phi$ is an isomorphism. I just don't know if it is ...