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 and prime elements

In my commutative algebra lecture notes it says: A non-zero element $p$ of a ring $R$ which is not a unit of $R$ is called a prime element if $p=ab$ implies $a$ is a unit or $b$ is a unit. Is this ...
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26 views

$\Bbb{Z}[X]/\langle f\rangle$ is a finitely generated $\Bbb{Z}$-module

Let $f$ be a monic polynomial in $\Bbb{Z}[X]$. Show that $\Bbb{Z}[X]/〈f〉$ is a finitely generated $\Bbb{Z}$-module. I don't even know how to start. If $g\in\Bbb{Z}[X]/〈f〉$, we are trying to find ...
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29 views

Proving that if the semigroup (A, *) is a group, then the relation is an equivalence relation.

I'm aware that posting exam questions is probably frowned upon, but this isn't homework, I think I'm genuinely misunderstanding some part of the algebra. The question is this: Throughout this ...
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26 views

Homomorphism between finite groups

I have to prove or disprove the following statement: If $\phi:G \rightarrow H$ is a homomorphism between finite groups, with non-trivial image (i.e. $\phi(G)\neq\{e_H\}$), then $\#G$ and $\#H$ ...
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36 views

$\Bbb{Q}$ is not a finitely generated $\Bbb{Z}$-module

I'm trying to show that $\Bbb{Q}$ is not a finitely generated $\Bbb{Z}$-module. Assume to the contrary that $$\Bbb{Q}=\Bbb{Z}\dfrac{a_1}{b_1}+...+\Bbb{Z}\dfrac{a_n}{b_n}$$ where $a_i,b_i\in\Bbb{Z}$. ...
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54 views

Does there exist a unital ring whose underlying abelian group is $\mathbb{Q}^*$?

Let $\mathbb{Q}^*$ be the group of units of the rational numbers. Does there exist a unital ring whose underlying additive group is $\mathbb{Q}^*$? I don't really have a gut feeling yea or nea. ...
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1answer
23 views

Isomorphism of quotient of direct sum modules

Let $M, N, M'$ and $N'$ be R-modules. If $M'$ and $N'$ are submodules of both $M$ and $N$ then is it true that \begin{equation} \frac{M}{M'} \oplus \frac{N}{N'} \cong \frac{M \oplus N}{M' \oplus N'} ...
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29 views

Homomorphism from a finitely generated module to a direct sum of modules

Let $R$ be a commutative ring with unit. If $M$ and $N_i$ are arbitrary $R$-modules, the module $\operatorname{Hom}_R(M,\bigoplus_{i\in I}N_i)$ is not isomorphic to $\bigoplus_{i\in ...
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1answer
32 views

judge if nilradical equals jacobson radical

judge if nilradical equals jacobson radical 1)a noetherian ring that is not a artin ring. 2)a local integral domain that is not a field. 3)a integral domain with only finite number of ...
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39 views

Evaluate a rational function at infinity

In the context of the Tate pairing, I would like to know that it means to `evaluate' an $\mathbb{F}_{q^k}$-rational function at $\infty$. For instance, the reduced Tate pairing is $e_n:G_1\times ...
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25 views

free group with 2 generators (two matrices)

Let $\alpha$ be complex number such that $| \alpha | \geq 1$.Show that $\left(\begin{array}{cc}1&0\\\alpha&1\end{array}\right) $ and ...
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1answer
18 views

Showing that Euclidean domain is UFD

Let $D$ be euclidean domain. We claim the following: $1)$ Every element of $D$ can be expressed as a product of irreducible elements $2)$ Every irreducible element of $D$ is a prime element. From ...
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1answer
41 views

Example of non noetherian ring and noetherian $\Bbb Z$-module

a non Noetherian ring that is a Noetherian $\Bbb Z$-module a Noetherian ring that is a non Noetherian $\Bbb Z$-module I have no idea in 1, and I'm not sure if $\mathbf{Q}$ is right for 2? ...
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36 views

How to prove homomorphism identity

I have problems to do the problem in http://www.math.helsinki.fi/kurssit/alggeom/h1.gif Let $k$ be a commutative ring and $f\in k[X_1,\ldots,X_n]$. Let $k^\prime,k^{\prime\prime}$ be commutative ...
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1answer
45 views

How can I set a Polynomial Algebra?

I was reading about Algebra over a field, and I see the definition of $\mathbb{K}\langle X \rangle$ as follows: Let $X \neq \emptyset$ a set, a word $w$ is an expression in the following way: ...
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1answer
42 views

An Infinite Cyclic Group has Exactly Two Generators: Is My Proof Correct?

I have completed a proof of this that I am inclined to believe is correct, or at least on the right track. I would like to ask if it is indeed correct, or if I need a nudge in the right direction. ...
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1answer
32 views

Subgroups and an union of orbits

I have to prove or disprove the following statement: If a group $G$ acts on a set $X$, then every subgroup $H$ of $G$ acts on the set $X$ as well, and every orbit of the action $G$ on $X$ is an ...
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49 views

Why field of fractions of $k[x_1,x_2,…]$ is Noetherian? [on hold]

the classical counterexample of a subring of a noetherian rings that is not noetherian is $k[x_1,x_2,...]$, which is not noetherian, but the field of fractions of $k[x_1,x_2,...]$ is, can anyone ...
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2answers
43 views

Group theory, the squares of G

We have a group $G$ with a subgroup $G_2$, which is defined by $G_2:=\{g^2|g \in G \}$. I have to prove that i) $G_2\triangleleft G$ ii) all elements of $G/G_2$ have order $\leq2$ iii)if ...
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66 views

Free finitely generated modules

Let $A$ be a ring and consider the free modules $A^{\oplus n}$, $A^{\oplus k}$, with $n,k\in \mathbb{N}$. Can $A^{\oplus n}$ be isomorphic to $A^{\oplus k}$ if $k\neq n$? Thanks in advance for the ...
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1answer
44 views

Finding a surjective homomorphism

I have to show that for all groups with $2007(=3^2\times223$) elements that there exists a surjective homomorphism to a group of 9 elements. Obviously a group with 2007 elements has a subgroup of ...
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1answer
36 views

if $A$ and $B$ are subnormal, then $A\cap B$ is subnormal [on hold]

A subgroup $X$ of a group $G$ is said to be subnormal if there exists a series $$X=X_0\subseteq X_1\subseteq\cdots\subseteq X_n=G$$ where each $X_i$ is normal in $X_{i+1}$. Prove that if $A$ and $B$ ...
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35 views

Finite non-abelian group $G$ containing a subgroup $H_0\neq\{e\}$ with property that $H_0\leq H$ for all subgroups $H\neq\{e\}$ [on hold]

Give an example of a finite non-abelian group $G$ containing a subgroup $H_0\neq\{e\}$ with property that $H_0\leq H$ for all subgroups $H\neq\{e\}$ of $G$ This question is from Herstein Topics in ...
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54 views

Prove that an ideal is not maximal

Ring $\mathbb Z[x],$ ideal is $(x)$. How to prove that this is NOT a maximal ideal? I can't imagine ideal, part of which would be $(x)$.
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Subgroups of $S_{n}$ of index $n$

We know that for $n\geq{5}$ any subgroup of $S_{n}$ of index $n$ is isomorphic to $S_{n-1}$. We know that by looking at the set of functions that fix a given $j$, we can obtain $n$ such subgroups ...
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1answer
17 views

$|\{ x\in X: g.x=x \space\space\space \forall g\in G \}| = |X|\space mod \space p$

Let $G$ be a p-group. $|G|=p^n$ for some n. Let X be a finite set so that $\,p\nmid |X|\,$, G acts upon X. Denote $A:= \{ x\in X: g.x=x \space\space\space \forall g\in G \}$ I am trying to show ...
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44 views

every ideal is contained in a maximal ideal

The statement is: In a commutative ring with 1, every proper ideal is contained in a maximal ideal. and we prove it using Zorn's lemma, that is, $I$ is an ideal, $P=\{I\subset A\mid A\text{ is ...
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75 views

inverse limit in the plane

What stuff can I say about inverse limits regarding the mapping of $[0, 1]$ onto $[0,1]$ given by $$f(x) = \left\{ \begin{array}{ll} 2x & \mbox{if } 0 \le x \le {1\over2}\\ 1 & \mbox{if ...
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31 views

Gröbner Basis and linear basis

Let $I$ be an ideal of a polynomial algebra $A$ with a Gröbner basis $G$. Suppose we know how to describe the leading terms of all elements in $G$, denoted by $\{i_1,\dots,i_k\}$, so that we can give ...
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any $2$-dimensional rep of a finite, non-abelian simple group is trivial

Let $G$ be a finite, non-abelian simple group. How would I go about proving that any $2$-dimensional representation of $G$ is trivial? If it helps, I know how to do it when we're considering ...
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1answer
41 views

Are the two ways of creating an $S^{-1}A$ algebra equivalent?

Let $f:A\to B$ be a ring homomorphism and $S$ be a multiplicative set, define $S^{-1}B$ to be $B\times S$ with equivalence relation $(b,s)\sim(b',s')$ iff $\exists t\in S$ such that $t(sb'-s'b) = 0$. ...
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42 views

MCS meet all prime ideals

let A be a commutative ring, is there any multiplicatively closed subset S (not containning 0), s.t. every prime ideal in A intercept S is not empty? My thinking is that there is 1-1 ...
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44 views

Irreducible polynomial and primes [duplicate]

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 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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97 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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94 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
66 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
46 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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Where does the proof for commutative rings break down in the non-commutative ring when showing only two ideals implies the ring is a field?

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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Does an arbitrary product of $f$ and $f^\dagger$ belong to a universal enveloping algebra of the Heisenberg algebra?

The Heisenberg algebra is essentially the canonical commutation relations (CCR) for bosons $[f,f^\dagger]=1$. $f$ is called an annihilation operator in physics ($f^\dagger$ creation operator). ...
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158 views

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

We know that torsion free plus finitely generated $\rightarrow$ free and that $\mathbf{Q}$ is torsion free is easy. But how to show $\mathbf{Q}$ 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 first image has the same symmetry group as the small embedded square - namely the dihedral group $D_4$ - with $8$ elements and generators ${x,y}$ with $x^4 = e, y^2 ...
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29 views

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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119 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
51 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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50 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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29 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
53 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.