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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If $I$ and $J$ are ideals of an $R$-algebra $A=I+J$ then $I\oplus J\simeq A\oplus (I\cap J)$?

Let $I$ and $J$ be two left ideals of an $R$-algebra $A$ such that $A=I+J$. Here $R$ is commutative ring with identity $1_R$. How can I show $$I\oplus J\simeq A\oplus (I\cap J)?$$ I've tried several ...
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Question about the deduction of the quotient ring $R/I$

Yesterday we deduced on class how quotient groups were deduced and well defined. Let $R$ be a ring and $I$ an ideal of $R$. My professor proved us that the multiplication operation $$R/I \times R/I ...
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Prove this enticing equation [on hold]

If $x=(a/b)^{2ab/(a^2-b^2)}$ I want to prove that $$( (ab)/(a^2+b^2) )(x^{a/b} + x ^ {b/a})=(a/b)^{(a^2+b^2)/(a^2 - b^2)}.$$
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3answers
35 views

Prove or disprove $\mathbb{Q}[x] /(x^5-3) \cong \mathbb{Q}[x] /(x^5-9)$

Want to prove or disprove this $\mathbb{Q}[x] /(x^5-3) \cong \mathbb{Q}[x] /(x^5-9)$ as communtative rings. I can show that $x^5-3$ and $x^5-9$ are irreducible in $\mathbb{Q}$, but I cannot go from ...
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0answers
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Number of equations needed to determine set of functions

Consider the set of all binary functions on $\{0,1\}$. An equation like $xy=yx$ determines the subset of all commutative binary functions. For any subset of the set of all binary functions on ...
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2answers
14 views

Is there another terminology to designate this?

Let $R$ be a principal ideal domain. Let $M$ be a finitely generated $R$-module. Then there exists a free $R$-submodule $F$ of $M$ such that $M=Tor(M)\oplus F$ and the ranks of such $F$'s are the ...
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1answer
12 views

Show that w-complete Posets and continuous aplications between them form a category

I'm really lost with this thing that looks innocent but just can't figure out... can you help me? Show that $\omega$-complete Posets and continuous functions between them form a category. Thank ...
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28 views

Polynomial Combinations in $F[x]$

Supposed $f(x), g(x) \in F[x]$ for some field $F$ are polynomials of degrees $m, n $ respectively. Moreover assume that they are relatively prime. By Euclidean algorithm I can find $a'(x), b'(x)$ such ...
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1answer
48 views

Difference between definitions of $p$-subgroup and Sylow $p$-subgroup

I'm reading Abstract algebra by Dummit and Foote and the following definitions are made: $1$. A group of order $p^{\alpha}$ for some $\alpha\geq1$ is called a $p$-group. Subgroups of $G$ which are ...
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3answers
34 views

Example of ideal generated by two elements

I have an easy example on my notes that I don't understand. My teacher said that in $\mathbb{Z}$, $(2,3)=2\mathbb{Z}+3\mathbb{Z}$ is a principal ideal, because $2\mathbb{Z}+3\mathbb{Z}=\mathbb{Z}$. ...
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41 views

Does every finite field have a subfield $\mathbb{Z}_p$?

It seems that in the answers for my exercises in the book, the book uses that every finite field, has a subfield $\mathbb{Z}_p$. Is this true? They seem to use it in the answer for one exercise. But ...
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1answer
19 views

If $\sigma\in S_p$ with $|\sigma| = p$, why is $\sigma$ a $p-$cycle, and why is $|\sigma^r| = p$?

If $\sigma\in S_p$ with $|\sigma| = p$, why is $\sigma$ a $p-$cycle, and why is $|\sigma^r| = p$ for each $r$, $1\leq r < p$? ($p$ is a prime) I guess I am just having a hard time understanding ...
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1answer
30 views

Example of a ring without unity that has a subring with unity?

I can't think of a ring without unity that has a subring with unity. There must be some element in the parent ring that doesn't work with the subring's identity, but I'm struggling to see how that ...
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19 views

Zorn's Lemma Application for Finding Maximal Submodule?

I came across the following exercise in my algebra textbook. Let $M$ be a left $R$-module ($R$ is ring with unity $1_R$) and let $N\subseteq M$ be a submodule such that $x\in M-N$ for some $x\in ...
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1answer
16 views

Prove that $R$ has ACCP if and only if every non-empty collection of principal ideals of $R$ has a maximal element.

Let $R$ be a commutative ring. Prove that $R$ has ACCP if and only if every non-empty collection of principal ideals of $R$ has a maximal element. Prove further that if $R$ is an integral domain and ...
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2answers
79 views

Prove that $x$ has order $5$.

let $ x \in G$ such that $(a^{-1})*(x^2)*(a) = x^3$ for some self inverse $a.$ Prove that $x$ has order $5.$ I don't know how to start this proof. Seems really difficult.
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2answers
54 views

Is every group of odd order isomorphic to a subgroup of $A_n$ for some $n$?

Is every group of odd order isomorphic to a subgroup of $A_n$ for some $n$? If not, what is a counterexample; if so, how can I prove it? Hints will be appreciated.
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2answers
32 views

The relation $\cong$ an equivalence relation on nonempty collection of groups $\mathcal{G}$

The following text is given in Dummit and Foote (pg-$37$) as example of isomorphism: For any group $G$, $G \cong G$. The identity map provides an obvious isomorphism but not, in general, the ...
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1answer
25 views

If $G$ is finite group that supersoluble then $G$ satisfy the maximal permutizer condition?

The 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 | \langle x \rangle H = H \langle x \rangle ...
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60 views

Does it make sense to talk about unit in a ring that does not contain 1?

Does it make sense to talk about unit in a ring that does not contain 1? I am intending to prove that if $r$ is irreducible and $r \nmid d$, then gcd(r, d) = 1. The definition of irreducibility ...
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21 views

covering finite Dimensional vector space

can a finite dimensional symplectic vector space over finite field be covered with mutually transversal Lagrangian planes(maximal Isotropical Subspaces )?
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21 views

What is the permutizer of the Sylow 3 subgroup in $S_4$ ? [on hold]

The 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 | \langle x \rangle H = H \langle x \rangle ...
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1answer
28 views

Algebraically closed field and it's characteristic

Ok so my question is motivated by the theory of Lie algebras, and seeing as I'm not that familiar with a lot of group theoretic notions, just the basics really, my question is as follows. What can be ...
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1answer
35 views

Representations of group $G=\mathbb{Z}/p\mathbb{Z} \oplus \mathbb{Z}/p\mathbb{Z} $

I need to find all in-equivalent irreducible representation of a group $G \equiv \mathbb{Z}/p\mathbb{Z} \oplus \mathbb{Z}/p\mathbb{Z} $ I know that $\mathbb{Z}/p\mathbb{Z}$ is a cyclic finite group. ...
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Analogue of Dirichlet $L$-function for $\mathbb{F}_q[T]$.

We consider an analogue of the Dirichlet $L$-function in $\mathbb{F}_q[T]$. Let $g \in \mathbb{F}_q[T]$, $g \neq 0$, let $\chi: (\mathbb{F}_q[T]/(g))^\times \to \mathbb{C}^\times$ be a homomorphism, ...
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2answers
56 views

Find all solutions of equation $x^{23}=5$ in $\Bbb Z_{23}$

I just found that $5$ is a solution by using Fermat's theorem. But, I am not sure whether there are more solutions and how I could find them...
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Proving that extension by radicals implies solvable group

I'm trying to understand the following excerpt from Fraleigh's A First Course In Abstract Algebra, Seventh Edition, pp. 472-473: 56.4 Theorem Let $F$ be a field of characteristic zero, and let ...
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Show that $a + b\sqrt{n}$ is an irreducible element of $\mathbb{Z}[\sqrt{n}]$

Let $n(\neq 0,1)$ be a square-free integer. Suppose that $|a^2 - nb^2|$ is a prime integer for $a,b \in \mathbb{Z}$. Show that $a + b\sqrt{n}$ is an irreducible element of $\mathbb{Z}[\sqrt{n}]$. ...
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Confusion between polynomial in field and factorization.

Consider $f(x)=x^3+3x+2$ in $\mathbb{Z}_5[x]$ and we can see that this polynomial is irreducible over $\mathbb{Z}_5[x]$ since it has no zeros in $\mathbb{Z}_5$. After I read this example and found ...
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1answer
28 views

Quotient of the ring of integers of a quadratic field by the ideal generated by a split integer prime.

I am wondering about primes $p$ in $\mathbb Z$ that are split in $\mathcal O_{K}$, $K=\mathbb Q(\sqrt d)$. Let $\omega=\sqrt d$ if $d \equiv 2,3 \mod 4$ and $\omega=\frac{1+\sqrt d}{2}$ if $d \equiv 1 ...
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What are the differences in mental skills required to master abstract algebra and analysis?? [on hold]

I had took undergraduate-level abstract algebra and analysis courses before. And I find I can do proofs in analysis faster than in abstract algebra. However some other students is opposite to me. I ...
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5answers
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$x^4 + 1$ reducible over $\mathbb{R}$… is this possible?

I am seeing this on a homework and am wondering if this is a typo. I am aware that $x^4 + 1$ is irreducible over $\mathbb{Q}$. I know the following: A polynomial being irreducible over some ring ...
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3answers
44 views

Show that the finite Abelian group is cyclic

Suppose that $G$ is a finite Abelian group that has exactly one subgroup for each divisor of |$G|$. Show that $G$ is cyclic. What I have so far: By the Fundamental Theorem of Finite Abelian Groups, ...
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49 views

How to prove the group of automorphisms of $S^1$ as a topological group is $\mathbb Z_2$?

The title basically says it all. How does one prove the group of automorphisms of $S^1$ (the unit circle in $\mathbb C$), as a topological group, is $\mathbb Z_2$? I was surprised not to find the ...
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0answers
29 views

What is the complexity of finding degree 5 polynomial integer solutions? [on hold]

Given a degree 5 polynomial with integer coefficients, its Galois group is generally unsolvable. But suppose that we restrict the polynomial space to the space of polynomials with only integer ...
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57 views

Definition of $\mathfrak{m}$-adic completion.

Let $V$ be a discrete valuation ring with the maximal ideal $\mathfrak{m}$ and let $T$ be a prime element of $V$. Assume that we have a subfield $k\subseteq V$ such that the induced map $ k \to ...
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How would you explain a quadratic field to a beginner?

How would you explain a quadratic field to a beginner? Eg. how did the subject first start? All the modern stuff they use to explain it makes it really confusing how one should think about it in more ...
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1answer
20 views

Quadratic function with positive integral coefficients problem

Here is the problem statement: Let $f(x)$ is a quadaratic expression with positive integral coefficients such that for every $\alpha, \beta\; \epsilon\; \Re$, $\beta>\alpha$, $\int_\alpha^\beta ...
2
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3answers
86 views

Is there an counterexample for the claim: if $A \bigoplus B\cong A\bigoplus C$ then $B\cong C$

Here A, B and C are R-modules. Is there an counterexample for the claim: if $A \bigoplus B\cong A\bigoplus C$ then $B\cong C$? And what if B and C are finitely generated?
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1answer
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Does $\alpha$ need to be transcendental over F?

In the book there is this exercise: Let E be an extension fiel of F, with $\alpha, \beta \in E$. Suppose $\alpha$ is transcendental over F but algebraic over $F(\beta)$. Show that $\beta$ is ...
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4answers
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Polynomial: Is there a theorem that can save my proof when $K$ doesn't include $\mathbb C$

Suppose $f(x),g(x)\in K[x]$ ($K$ a number field), let $f(x)=x^{3m}+x^{3n+1}+x^{3p+2}$, where $m,n,p\in\mathbb N$, and let $g(x)=x^2+x+1$, prove: $$g(x)\mid f(x)$$ I think this problem is not ...
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1answer
47 views

Prove that $f$ is an onto function and a homomorphism function from $(\mathbb{Z} \times \mathbb{Z}, \oplus)$ to $(\mathbb{Z}, +)$

I have a lot of issues trying to figure out this problem. Any advice? Consider the two groups $(\mathbb{Z} \times \mathbb{Z}, \oplus)$ and $(\mathbb{Z}, +)$, where $(a,b) \oplus (c,d) = (a + c, b + ...
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1answer
50 views

Why is this map injective?

On the page we find the following: $ \phi : Z ( P_1 , \dots , P_r ) \to \mathrm {Spm} (K[ X_1 , \dots , k_n ] / \sqrt{ ( P_1 , \dots , P_r )}) $ defined by $ \phi ( ( a_1 , \dots , a_n ) = \pi ( ( ...
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2answers
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Prove that $r$ is irreducible if and only if every divisor of $r$ is either a unit or an associate of $r$.

Let $R$ be an integral domain, and let $r \in R$ be a non-zero non-unit. Prove that $r$ is irreducible if and only if every divisor of $r$ is either a unit or an associate of $r$. Proof. ...
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70 views

What are all the integral domains that are not division rings?

A commutative division ring is an integral domain. But what are all the integral domains that are not division rings? The examples I currently know are the following: $\mathbb{Z}$, $\mathbb{Z}[i]$, ...
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1answer
36 views

Prove that $2$ is irreducible in $\mathbb{Z}[\sqrt{n}]$ if $n$ has a prime factor congruent to $5$ modulo $8$.

Prove that $2$ is irreducible in $\mathbb{Z}[\sqrt{n}]$ if $n$ has a prime factor congruent to $5$ modulo $8$. I know that if $x^2 \equiv \pm2 \pmod p$, where $p$ is a prime, has no solution if ...
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2answers
39 views

Principal ideal domain that is not Euclidean domain.

An example of such ring is $\mathbb Z [(1+\sqrt{-19})/2]$. But in the proof, there is something hard to understand. $ax+by+cz=1$ where $a,b,c,x,y,z$ are integers and $c>1$. Then why can we write ...
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2answers
40 views

Proof involving homomorphism - Abstract Algebra

I will greatly appreciate details here. Theorem: Let $\phi: G \to H$ be a group homomorphism: - $$\text{if}\space\space L \leq H, \space\space \text{then} \space\space \phi^{-1}(L) \leq G $$ ...
3
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0answers
34 views

Proof that $\mathbb{Z}[\sqrt{3}]$ is a Euclidean Domain.

Proof that $\mathbb{Z}[\sqrt{3}]$ is a Euclidean Domain. Proof that $\mathbb Z[\sqrt{3}]$ is a Euclidean Domain Is it possible to solve this question without using $\mathbb{Q}[\sqrt{3}]$ restricted ...
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1answer
22 views

What is the free module of this one?

Let $R=\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z}$ and $A=\{(1,0)\}$. Then, what is the free $R$-module $F(A)$? (Here the module action is mutiplying componentwise. ) I tried spanning $A$ by ...