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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Irreducible polynomial and primes

Let $n$ be a prime number. How can I show that the polynomial $f(x) = x^{n-1} + x^{n-2} + x^{n-3}+ \cdots + x+ 1$ is irreducible over any finite field?
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What are the notations $k^{\prime n}$ and $\varphi^n$ in algebra?

I would like to understand what do the following problem says: Let $k$ be a commutative ring and $f\in k[X_1,\ldots,X_n]$. Let $k^\prime,k^{\prime\prime}$ be commutative $k$-algebras and ...
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1answer
35 views

Group of order $p^2$ is commutative with prime $p$

Please help me on this one: Let $p$ be a prime number, show that each group of order $p^2$ is commutative. If you do not mind at all, could you please not give me the elegant explanation, but ...
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85 views

Why not define $|v| = -1$? [on hold]

I was wondering why if we have $i^2 = -1$, why not have a "number" $v$ such that $|v| = -1$? Does anything interesting arise from considering this system? The only thing I could come up with was: ...
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1answer
58 views

Why $ \mathbb{Z}[x]$ is not Principal Ideal Domain [duplicate]

$ \mathbf{Z}[x]$ is not PID. we know $\mathbb Z$ is a Unique Factorization Domain, so $\mathbb Z[x]$ is UFD, but why isn't it PID (since I think $\mathbb Z$ is PID)?
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Group Transitive Action's Effect on Stabilizers's Conjugacy

I am looking for guidance for two problems on group action, one of them is here and the other one has just been posted earlier: Assume that $G$ operates on a set $\Omega.$ Show that, if $G$ acts ...
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2answers
44 views

Minimal subring of complex numbers

Let $\alpha$ be a root of $X^3+X^2-2X+8$. My question is if $\mathbb Z\left[\alpha,\frac{\alpha+\alpha^2}{2}\right]=\{a+b\alpha +c\frac{\alpha+\alpha^2}{2}:a,b,c\in\mathbb Z\}$? Thank you all.
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Intersect of Stabilizers is a Normal Group

I am looking for guidance for two problems on group action, one of them is here and the other one will be posted in another page: Assume that $G$ operates on a set $\Omega.$ Show that ...
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3answers
56 views

Why isn't a (noncommutative) ring with only trivial ideals a division ring?

We know in a commutative ring, if the only ideals are trivial and the whole ring, then the ring is a field, which is proved by every ideal is contained in a maximal ideal, which is proved by Zorn's ...
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105 views

how to show $\mathbf{Q} $ is not free

we know torsion free plus finitely generated $\rightarrow$ free and that $\mathbf{Q}$ is torsion free is easy. But how to show Q is not finitely generated? and not free?
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Which convex $2n$-gons have symmetry group $D_n$ instead of $D_{2n}$?

The equilateral octagon $M$ in the image has the same symmetry group as the embedded square - namely the dihedral $D_4$ - with $8$ elements and generators $[x,y]$ with $x^4 = e, y^2 = e $ and $yxy = ...
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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
96 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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49 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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27 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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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
48 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
33 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
25 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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41 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
23 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)= ...
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A question on 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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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
67 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
83 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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1answer
47 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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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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79 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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2answers
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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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1answer
63 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
45 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
77 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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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
62 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
45 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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2answers
70 views

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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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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73 views

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
34 views

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 = ...
2
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2answers
47 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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37 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 ...