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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Is there any automorphism $f$ that satisfies these requirements? [on hold]

Suppose that $\Bbb R $ is the set of real numbers. Is there any automorphism $f$ from $(\mathbb R,+)$ to $(\mathbb R,+)$ of the following form? $$f(x)=kx, \quad k \neq 0,1 , \quad k \in \Bbb R$$
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1answer
78 views

If $G/G'$ is finite, then $|Z(G)| < \infty$

Let $G$ be an infinite group. Suppose that $G/G^{\prime}$ is a finite group, where $$G^{\prime}= \left\langle xyx^{-1}y^{-1}\ \middle\vert\ x,y \in G\right\rangle.$$ Prove that $|Z(G)| < \infty$.
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1answer
44 views

Number of group actions [on hold]

In how many ways can the group $\mathbb{Z}_5$ act on $\{1,2,3,4,5\}$.
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20 views

Use Pohlig-Hellman to solve discrete log

We have $$7^x = 166 \pmod{433}$$ I need to find $x$ using the Pohlig-Hellman algorithm.
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22 views

Density of odd vs even group orders that are not forced to be simple by Sylow's Theorem

In Dummit & Foote's Abstract Algebra text, I've just solved the following two Exercises: Write a computer program which (i) gives each odd number $n<10,000$ that is not a power of a ...
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1answer
47 views

How to find fast modular exponentiation?

I need to find $$5448^{5^3} \pmod{11251}$$ and $$6909^{5^3} \pmod{11251}$$ fast? I couldn't calculate it with normal tricks.
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1answer
27 views

Define a projection homomorphism and find the kernel

I was given the projection homomorphism $Z_4 \times Z_3 \to Z_3$ and asked to find it and come up with the kernel. I came up with $\phi(x,y)= x$ such that $x \in Z_4$ and $y \in Z_3$ ...
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1answer
23 views

Order of elements in finite fields

Show that 5448 has order 5^4 in Z/11251Z. How do I show this quickly, I know that all elements must have an order that divides order(Z/11251) according to lagrange's theorem but how do i pinpoint that ...
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1answer
20 views

Does the set operation define a binary operation on G?

Consider the set G = {0,{1},{2},{1,2}}. Does the set operation intersection de fine a binary operation on G? Does the set operation union de fine a binary operation on G? Is < G,(union) > a group? ...
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2answers
39 views

Show that $f _ a $ is a Homomorphism

For a fixed element $a$ is a group $G$, define $$f _ a (x) = a ^ {−1} xa , x \in G$$ Show that $f _ a $ is a Homomorphism. I know that to show that a mapping $f:G \rightarrow G'$, Where $G$ and $G'$ ...
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1answer
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$M$ noetherian, $f$ endomorphism of $M$, $\operatorname{coker}f$ has finite length, then $\operatorname{coker}f^n$ and $\ker f^n$ have finite length.

Let $M$ be noetherian and let $f$ be an endomorphism of $M$. Suppose that $\operatorname{coker}f$ has finite length. Prove that both $\operatorname{coker}f^n$ and $\ker f^n$ have finite length ...
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1answer
18 views

NTRU cryptosystem

For the NTRU cryptosystem (as described here http://en.wikipedia.org/wiki/NTRUEncrypt), why is it really easy for Eve to decrypt if $p$ divides $q$. My answer was that when Eve sees $e(x)= ...
2
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1answer
21 views

A question of Weyl algebra $A_1$

This question is from A Primer of Algebraic D-Modules by S.C. Coutinho, Ex 2.4.10. In the following $A_1$ means the Weyl algebra generated by $x$ and $d$. Let $f: A_1^2\to A_1$ be the map defined ...
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4answers
104 views

Prove that $\mathbb{Z}$ and $\mathbb{Q}$ are not isomorphic as groups

I'm working on the following problem: Prove that $\mathbb{Z}$ and $\mathbb{Q}$ are not isomorphic as groups. Here is my attempt at a solution: If $\mathbb{Z} \cong \mathbb{Q}$, then there must ...
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2answers
66 views

Number of subgroup of order $p^2$ in $\mathbb{Z}_{p^{2}} \times \mathbb{Z}_{p^{2}}$

Let $G = \mathbb{Z}_{p^{2}} \times \mathbb{Z}_{p^{2}}$. How many subgroups does $G$ has of order $p^2$? I know there are only 2 cases of the subgroup H, H can be isomorphic to $Z_{p^{2}}$ or $ ...
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1answer
78 views

Is $(G,*)$ commutative? [on hold]

$(G,*)$ is a group and for some three consecutive integers $i=j,j+1,j+2$, it satisfies $(a*b)^i=a^i*b^i$ for every $a,b\in G$. Is $(G,*)$ commutative?
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43 views

Finite fields and arithmetic

For every prime number $p$ and every positive integer $k$, there is a field with exactly $p^k$ elements. When $k=1$, it's just the integers$\bmod p^k$, and when $k>1$, it's not. So if I want the ...
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0answers
49 views

Multiplying rings: [on hold]

How would you multiply two rings? Given the ring Znx Zm, would you get out Znm, or is this not the way that you do it? I am asking because I know that for in order to find the characteristic of a ...
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2answers
78 views

List the elements of G

So we are asked to write out the elements of G and H where G= $ \mathbb{Z}/ <20> $ and H = $<4, 20>$ . I understand how to do H and I got: {$0 + <20>, 4 + <20> , 8 + ...
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Does a subring of $\mathbb{Z}$ need to be closed under multiplication?

I know that the ring $\mathbb{Z}$ has the binary operation under addition, and when we are trying to identify whether a given ring is a subring of $\mathbb{Z}$, the subring must contain: the identity, ...
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62 views

An abelian subgroup of symmetric group

PROBLEM: Let $G$ be an abelian subgroup of the symmetric group $S_n$ and $p_1, . . . , p_k$ be all prime divisors of $|G|$. Prove that $n≥p_1 +···+p_k$. QUESTION: How do you solve this problem. I've ...
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1answer
41 views

If R is an integral domain disprove the RxR is an integral domain? [duplicate]

I am trying to prove that given R (an integral domain) it is not true that then RxR is an integral domain: We know that for the ring Zp for any prime p, Zp is an integral domain because it is a ring ...
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1answer
74 views

A question about field extension: Zariski's lemma

Suppose $E$ is a field extension of $F$ and there exists $\alpha_1,\alpha_2,\ldots,\alpha_n\in E$ such that $E=F[\alpha_1,\alpha_2,\ldots,\alpha_n]$, then the field extension $E/F$ is algebraic. Is ...
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1answer
36 views

Groups of Order 12 aren't Simple

Suppose $G$ is a group of order $12=2^2*3$. Let $n_p$ denote the number of Sylow p subgroups. Then $n_2$ is 1 or 3 and $n_3$ is 1 or 4. I want to show that one of them is one since if that is the case ...
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3answers
34 views

Is is true that $R[x,y]/(x-y^2) \cong R[y]$?

I have a ring $R$ and I want to prove (or disprove) that $R[x,y]/(x-y^2) \cong R[y]$. My idea is to define a ring homomorphism $\phi$ such that $\phi$ is the identity on $R$ and such that $\phi(y) = ...
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3answers
60 views

Injective Ring Homomorphism

I seem to be having the wrong impression of what $p$ stands for; is $p(x)=x(x+1)(x+2)$ or is it something else? Clarification would be appreciated so that I can complete the lemma below. Consider ...
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1answer
43 views

Group actions: Why do we place the condition that $S$ be finite in the following theorem?

Theorem. Let $G$ be a group, $S$ be a $G$-set, and $S$ be finite, then $$|S|= \sum_{a \in A} [G : G_a],$$ where $A$ is a subset of $S$containing exactly one element from each orbit $[a]$. Here, $G_a$ ...
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How to factor polynomials in $\mathbb{Z}_n$?

How to factor a certain polynomial over $Zn$. for example factor the following polynomial into irreducible polynomials in $Z5$: $X^3+X^2+X-1$ or factor the following polynomial into irreducible ...
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1answer
23 views

Prove algebraic closure

Let $L/K$ be a field extension such that $L$ is algebraically closed. Show that $\{a\in L\mid[K(a):K]\lt\infty\}$ defines an algebraic closure of $K$. So this is the set of minimal polynomials ...
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Ramification group of valuations - need terminology

I am lost and need some terminology (also hopefully sources). Let $L/K$ be a Galois extension, and $w$ be a valuation of a $L$, lying above a valuation $v$ of $K$. Notice that I do not suppose that ...
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0answers
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Please show that $W$ is a normal subspace of $\sigma$. [on hold]

Suppose that $\sigma$ is a symmetry transformation, $V$ is a space, and $W$ is a subspace of $V$. Please show that $W$ is a normal subspace of $\sigma$.
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Solution verification: proving that $2$ is not prime in $\mathbb{Z}[\sqrt{-3}]$

I just took my final exam for abstract algebra and have this problem stuck in my head. Prove that $2$ is irreducible in $\mathbb{Z}[\sqrt{-3}]$ but not prime. My Solution: Proving that it is ...
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1answer
16 views

Prove center of a group is a subgroup using one-step subgroup test

I'm not sure if this is correct. It doesn't seem so. If $a,b \in C$, then we must show $ab^{-1} \in C$. $$ab^{-1}x=axb^{-1}=xab^{-1}$$ This doesn't seem correct. I've seen two-step subgroup tests ...
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1answer
22 views

Given transpositions, can you express the permutation in matrix form?

I know that if you are given the matrix itself or the disjoint cycles, you can easily express the permutation as a product of transpositions, but if you are only given the transpositions, can you go ...
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Galois group of the splitting field of the polynomial $x^5 - 2$ over $\mathbb Q$

We know that the splitting field of $x^5 - 2 $ over $\mathbb Q$ is $\mathbb Q(2^{1/5}, \rho)$, where $\rho$ is a fifth root of unity. Therefore, $\left[\mathbb Q(2^{1/5} , \rho) : \mathbb Q \right] = ...
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1answer
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How do i find maximum order of element in $S_{10}$ Group

Question is to find maximum possible order of an element in $S_{10}$ Group . Please someone help me through this .How to think of this intuitively .Thanks
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3answers
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Confused about a solution: proving every prime ideal is maximal

I'm looking at this solution to this problem: I'm getting thrown off by the special case where $n = 2$. If $n = 2$, why must it be that $x = 1$? All that we then know is that $x^2 = 1$ or that $x = ...
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2answers
44 views

Show that polynomial is reducible

Show that $p(x)$ = $x^3 + 3x^2 + 2x + 4$ is reducible in $\mathbb{Z}$$/$$7$$\mathbb{Z}$ Is the approach for this to factor it and then find a root? I'm a little confused on how to start. Any tips ...
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2answers
36 views

For any $p,q\in\mathbb{Z}[i]$, $\mathrm{N}(\gcd(p, q))$ must divide $\gcd(\mathrm{N}(p), \mathrm{N}(q))$

I'm studying for my final exam (abstract algebra) and am looking at an example where our professor was trying to compute the GCD of two elements of $\mathbb{Z}[i]$. Rather than directly applying the ...
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1answer
30 views

Show that $H$ is transitive on the set $G$.

Let $G$ be a group and let a be a fixed element of $G$. The map $\lambda_{a}: G \to G$, given by $\lambda_{a}(g) = ag$ for all $g \in G$, is a permutation of the set $G$. Note $H = \{\lambda_{a} ...
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1answer
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Irreducible elements and unique factorization domain

Let $P=\{\frac{a}{3^n} : a \in \mathbb{Z}, n \in \mathbb{N}\}$. a) Which elements are irreducible in $P$: 4, 5, 6, 9, 10, 15? b) Find out, which one of rings: $ P$, $\mathbb{Z}[i\sqrt{5}]$, $P[x]$ ...
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1answer
80 views

Is my understanding for group algebra correct?

Let $G$ be a group, $k$ be a commutative unital ring. Consider $\mathbf{Alg}_k^{inv}$ the category of unital $k$-algebras whose multiplicative semigroup is a group. Then there is a forgetful functor ...
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Product of numbers $\pm\sqrt{1}\pm\sqrt{2}\pm\cdots\pm\sqrt{n}$ is integer

Prove that the product of the $2^n$ numbers $\pm\sqrt{1}\pm\sqrt{2}\pm\cdots\pm\sqrt{n}$ is an integer. I want to consider the polynomial $P(x)=(x-a_1)(x-a_2)\cdots(x-a_{2^n})$, where the $a_i$'s are ...
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2answers
40 views

Gallian: is it true that the well-ordering principle can't be proven from properties of arithmetic?

I just started reading Gallian's Abstract algebra and on page 3 it says "An important property of the integers...is the so-called Well Ordering Principle. Since this property cannot be proved from the ...
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1answer
25 views

Showing that an epimorphism of an ideal is again an ideal

Let $R, S$ be commutative rings, $f : R \rightarrow S$ an epimorphism, I an ideal of R. Show that $f(I)$ is an ideal of $S$. As far as I understand, I need to show 4 things: 1) $0_s \in f(I)$ ...
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1answer
67 views

Is ideal prime or maximal? [on hold]

Find, whether or not given ideal of $\mathbb{Z}[x]$ ring is prime or maximal and describe the quotient ring : a) $J_1 = (x-5)$ b) $J_2 = (3, x+5)$. How can I do that?
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1answer
26 views

Example of finite ring with a non principal ideal

I would like to have an example (or a class of examples) of a finite ring (with unit, not necessarily commutative) that is not a principal ideal ring (if possible an example of a local ring and of ...
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1answer
107 views

What's the algebraic closure of the quaternions?

This post is a sequel of: Is the set of quaternions $\mathbb{H}$ algebraically closed? This answer shows that: 1. $\mathbb{H}$ is algebraically closed for the polynomials of the form $\sum a_r ...
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34 views

Determine $\operatorname{Hom}(\mathbb{Z},\mathbb{Z}/n)$ and $\operatorname{Hom}(\mathbb{Z}/n, \mathbb{Z})$. [on hold]

Determine $\operatorname{Hom}(\mathbb{Z},\mathbb{Z}/n)$ and $\operatorname{Hom}(\mathbb{Z}/n, \mathbb{Z})$, where $n$ is positive integer (as $\mathbb{Z}$-module). I can only find a ...
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1answer
31 views

Given the permutation, which symmetric group does it belong to?

Given the permutation (1, 2, 4)(3, 5, 6) is it clear which symmetric group this permutation belongs to? Explain. so from here I got: 1 2 3 4 5 6 2 4 5 1 6 3 and that is an element of S6 Im ...