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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How to prove an element is a unit if and only if the norm is

In the ring $\mathbb{Z}[\sqrt{2}]$, how do I prove that an element $\alpha$ is a unit if and only if $N(\alpha) = 1$? We are told that $N(a+b\sqrt{2}) = a^2-2b^2$. I've shown that ...
2
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1answer
45 views

Why do we have a basis?

A corollary that is in my book that I think is relevant to my question is: If E is an extension field of F, $\alpha \in E$ is algebraic over F, and $\beta \in F(\alpha)$, then $\deg(\beta,F)$ ...
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1answer
15 views

Cancellation of finitely generated modules over a PID

Suppose that $A=\mathbb R[x]$, $D = A/⟨x^2+1⟩⊕A^2$. $B$ and $C$ are finitely generated $A$-modules. Suppose that $D⊕B \cong D ⊕ C$. How to show $B\cong C$? I tried to solve it but I have no clue, ...
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2answers
22 views

Why is a submodule of a free module over a PID is free?

Rotman - Advanced modern algebra p.650 Theorem 9.8 Let $R$ be a PID and $M$ be a free $R$-module and $N$ be an $R$-submodule of $M$. Let $\beta$ be an $R$-basis for $M$ and well-order it. Now, ...
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0answers
12 views

L-module definition

I have the following definition of an L-module We say that V is an L-module if there is a k-bilinear mapping L × V → V sending a pair (x, v) ∈ L × V to x.v ∈ V such that [x, y].v = x.(y.v) − y.(x.v) ...
7
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58 views

Proving that the number of integer solutions of $x^2-Ny^2=1$ is infinite

I am trying to prove that the number of integer solutions of $x^2-Ny^2=1$ is infinite whenever N is a squarefree integer. For this I define norm of $a+b\sqrt N=a^2-Nb^2$. Now I prove that $a+b \sqrt ...
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0answers
14 views

Discriminant of a polynomial modulo a prime

If $p$ is a prime and divides the discriminant of an irreducible polynomial $f(x)=x^{n}+a_{n-1}x^{n-1}+\cdots+a_1x+a_0\in \mathbb{Z}[x]$ why is then $disc(f(x)\bmod p)=0$? I know that the ...
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2answers
23 views

Factor Ring question and finding maximal ideals of $\mathbb{Z}\times\mathbb{Z}$

What is the maximal ideal of $\mathbb{Z}\times\mathbb{Z}$? I think since $(\mathbb{Z}\times\mathbb{Z})/(\{0\}\times\mathbb{Z})$ is isomorphic to $\mathbb{Z}$, it seems like that ...
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0answers
11 views

Show that the set $\{[x_i,x_j]; i>j\}$ and $\{[x_i,x_j,x_k], i>j \leq k\}$ are linearly independent. [on hold]

Show that the set $\{[x_i,x_j]; i>j\}$ and $\{[x_i,x_j,x_k], i>j \leq k\}$ are linearly independent. I need to prove it to show that it is based on certain sets, but I can not prove, I believe ...
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19 views

Ribbon and colours [on hold]

A ribbon is composed from 9 square fabric pieces (i.e. is $1\times9$ rectangle). How many different ribbons can be made if there are fabrics of two colors and $5$ cells should be red and $4$ cells ...
0
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0answers
28 views

Need Help Understanding Why Proof Shows Set is not a Ring

I am having trouble reading this somewhat "slick" proof. Maybe it's not as slick as I think it is though, and I'm missing something here. So, I understand everything that is being done until the last ...
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2answers
19 views

question regarding group permutation computation

In S6, let $\alpha=(135)(156)(135)$ how do I compute $\alpha^{24}$? I'm given the hint that I first have to express alpha as a product of disjoint cycles which I computed as (15)(36) and then I have ...
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0answers
14 views

explicitly splitting the hamilton quaternions over local fields

For simplicity, lets first consider the hamilton quaternions $$ H = \left(\frac{-1,-1}{\mathbb{Q}}\right)$$ This is the central division algebra over $\mathbb{Q}$ with $\mathbb{Q}$-basis given by ...
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1answer
12 views

Show that $B^{(1)} = 0$ and $B^{(2)}$ have basis $\{[x_i,x_j]; i>j\}$

Definition: A polynomial $f \in K \langle X \rangle$ is called a proper polynomial, if it is a linear combination of products of commutators: $$f(x_1,x_2,...x_n) = \sum ...
2
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2answers
22 views

Maximal ideals and prime ideals of $\mathbb{Z}/2 \times \mathbb{Z}/2$?

I think there are 3 ideal and maximal primes. $<(0,1)>$ since factor group over $<(0,1)>$ is isomorphic to $\mathbb{Z}/2$, which is field and integral domain. And same reason for ...
2
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3answers
37 views

$\mathbb Q [\sqrt{2} i]$ contains neither $\sqrt[4]{2}$ nor $\sqrt{2}$

I want to prove that $x^4-2$ is irreducible over $\mathbb Q [\sqrt{2} i]$. In order to verify it has no linear factors and quadratic factors, I need to show $\mathbb Q [\sqrt{2} i]$ contains neither ...
1
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1answer
25 views

What is the difference between these two conditions $J = \{az \mid a \in R\}$ and $ I = \{a \in R \mid az \in J\}$

Please consider these two questions: Let $R$ be a ring and $z \in R$, which is fixed. Let, $J = \{az \mid a \in R\}$. Prove that $J$ is a left ideal of $R$. Skipping the subtraction part, this is ...
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23 views

Polya theorem and necklaces

How many $10-$ bead necklaces can you make out of $2$ red bead, $3$ blue beads and $5$ white beads $|G| = |D_{10}| = 20 $
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0answers
21 views

Ramification group - do you know/can produce a simple proof to this?

Let $(K,v)$ be a valuation field, $L$ a finite extension of $K$, and $w$ a valuation of $L$ above $v$. The ramification group of $w$ in $L$ is the subgroup of ${\rm Gal}(L/K)$ of all automorphisms ...
3
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0answers
26 views

Inner automorphisms as the kernel of a homomorphism

By a straightforward computation, it is not hard to show that the set $\operatorname{Inn}(G)$ of the inner automorphisms of a group $G$ is a normal subgroup of $\operatorname{Aut}(G)$, see for example ...
1
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1answer
41 views

Up to isomorphism

My notes state that if we only consider groups of size $n$ 'up to isomorphism', the elements of our group are a fixed set= {$a_1,..,a_n$}. I get that groups with the same structure but different ...
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3answers
40 views

Show if $2$ and $1+\sqrt{-5}$ belong to the same principal ideal $I$ of $\mathbb{Z}[\sqrt{-5}]$ then $I=\mathbb{Z}[\sqrt{-5}]$.

Show if $2$ and $1+\sqrt{-5}$ belong to the same principal ideal $I$ of $\mathbb{Z}[\sqrt{-5}]$ then $I=\mathbb{Z}[\sqrt{-5}]$. I have proved so far that 2 and $1+\sqrt{-5}$ is irreducible and ...
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1answer
19 views

On the minimal set of generators of ideals in $\mathbb{C}[x,y]$.

I am trying to do exercise 2.6 of Hassett's "Introduction to algebraic geometry": i) Give an example of a monomial ideal $I\subseteq\mathbb{C}[x,y]$ with a minimal set of generators consisting of ...
2
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1answer
43 views

relation between vector space and torsion free module

Can you please help me to prove this. Let $R$ be a domain, $A$ be an $R$-module, and $Q=Frac(R)$. Then a module $A$ is a vector space over $Q$ if and only if it is torsion-free and divisible. ...
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3answers
28 views

Why is it the smallest subfield containing F and $\alpha$?

Please take a look at the sentence in red: I understand that $\phi_\alpha[F[x]]$, is a subfield which contains $\alpha$, and F(we just need to evaluate $\phi_\alpha$ at the appropriate values). But ...
2
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1answer
21 views

integral domains and field of fractions

I've read about integral domains and their induced fields of fractions. For an integral domain $R$ its field of fractions $K$ is the "smallest" field that includes $R$, i.e. there is an injective map ...
2
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0answers
24 views

How to prove the cubic formula without root extraction

I'm trying to prove the cubic formula, in the following form: Given a field $F$ and $x,p,q\in F$, define $m=\frac p3$ and $n=\frac q2$, and suppose also that $\gamma,\tau$ are given such that ...
3
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1answer
33 views

Deducing that the inverse of a permutation matrix is its transpose

I would like to verify that my proof below is sound. Let $A\in P$ where $P$ is the set of all permutation matrices (only one 1 in each row and column). Also, let $(A)_{ij}$ denote the entry of $A$ in ...
0
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2answers
28 views

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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0answers
16 views

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 ...
3
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3answers
43 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
19 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
18 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 ...
4
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2answers
32 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 ...
3
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1answer
56 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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3answers
46 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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2answers
40 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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0answers
21 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 ...
2
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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 ...
9
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2answers
89 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
57 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
35 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 ...
2
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1answer
28 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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2answers
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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0answers
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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24 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 ...
0
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1answer
29 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 ...