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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Isn't reflexivity and symmetry implied in equivalence relations?

It looks like for all "nice" sets, the set $S\times S$ will have symmetry and reflexivity by default. The tough part is usually showing transitivity. However, are there any non-empty sets such that ...
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Groups and Subgroups elements

Let $G$ be a cyclic group order $12$ with $G=\left<a\right>$. Let $H=\left<a^3\right>$. List the elements of $H$ and find the cosets. I am lost as to what the elements of $H$ would be. ...
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To show that any ring of order $15$ is commutative.

To show that any ring of order $15$ is commutative. I am stuck with this proof. Please help. $15$ can be written as product of two primes $3$ and $5$.
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34 views

a question about abstract algebra(group theory)

(a)What are the fi nite abelian groups of order 100 up to isomorphism? (b). Say G is a fi nite abelian group of order 100 which contains an element of order 20 and no element with larger order. Then ...
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Prove there generally is no isomorphism between $R[x]/(x^2-a)$ and $R^2$

I have a ring $\mathbf R =(R, +, -, ., 0, 1)$ (note that there is no invers for multiplication, $R$ is not $\mathbb R$, it is any set for the given algebra). How do you prove that the following does ...
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25 views

Binomial of Mersenne prime power.

Let $f(x)$ be irreducible over $\mathbb{Z}_2$ of degree $p$, where $p$ is prime. Let $2^p-1$ be a Mersenne prime. I have to show \begin{equation*} f(x) \mid (x^{2^p-1}-1). \end{equation*} I am ...
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11 views

Induced character and center of groups

If $Z$ is a subgroup of the center of G and $|G:Z|=m$, then $\chi^*(g)=m\chi(g)$ if $g\in Z$ where $\chi$ is a character of $Z$ and $\chi^*$ is the induced character of $G$. Let $\phi$ be the ...
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19 views

Reference for Affine Spaces

I recently started reading Arnold's Mathematical Methods of Classical Mechanics (Second Edition). On pg. 4 Arnold writes: Affine $n$-dimensional space $A^n$ is distinguished from $\mathbb R^n$ in ...
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21 views

Universal property of natural number semi-ring

I asked a question similar to the one I am about to ask, and I think I got a satisfactory answer. However, this time I have some more specific question. Let a semiring $(R,+,\times)$ be an algebraic ...
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2answers
22 views

Product of linear factors

I have a polynomial $h=T^5+6T^4+6T^3+T+2$ in ring $\mathbb{F_7}[T]$. I should write it as a product of linear factors. So $h=(T+1)^2 (T^3+4T^2-3T+2)$. But $-3$ in $\mathbb{F_7}$ is $4$, so that ...
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18 views

Prove that the map $\theta(f(\alpha))=f^\sigma(\beta)$ is injective

I'm reading Lang's algebra chapter about Field theory and Galois theory. There is a theorem that says: Let $k$ be a field, $E$ an algebraic extension of $k$, and $\sigma:k\to L$ an embedding ...
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30 views

Find $\alpha$ s.t. $\mathbb{Q}(i,\sqrt[3]{2})$ is $\mathbb{Q}(\alpha)$

I want to find $\alpha$ s.t. $\mathbb{Q}(i,\sqrt[3]{2})$ is $\mathbb{Q}(\alpha)$, but i'm not sure how to do that. $i^2 \in \mathbb{Q}$ and $\sqrt(3)^3 \in \mathbb{Q}$ and $2$ and $3$ are coprime ...
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pls help me solve this math problem as fast as possile [on hold]

Imagine an alien species with 6 possible base pairs for its DNA. How many base pairs would it have to have to store about the same amount of information as the human genome?
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If $k|n, k \geq 2$, then $D_{n}$ has a subgroup isomorphic to $D_{k}$

Restatement of question: If $k|n, k \geq 2$, then the group $D_{n}$ has a subgroup isomorphic to the group $D_{k}$. My attempt at proving the result stated: Let us say that $D_{n}= \{1, \sigma, ...
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15 views

separable polynomial $ \bmod p$ (definition)

Given a polynomial $ f(x) \in K[x]$, where $K$ is a number field, we say that $f$ is separable if all its roots are distinct in an algebraic closure of $K$. Question: What does it mean a ...
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If I know $AB$, how can I calculate $BA$?

Let $A∈\mathscr{M}_{3×2}(\mathbb{R})$ and $B∈\mathscr{M}_{2\times3}(\mathbb{R})$ be matrices satisfying $AB =\begin{bmatrix} 8 &2 &−2\\ 2 &5 &4\\ −2 &4& 5 \end{bmatrix}$. ...
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Let $D$ be an integral Domain. Show that $\langle r\rangle = \langle s\rangle$ then $s = ur$ for some unit $u \in D$

My basic question is about the notation- in regards to $\langle r\rangle$ and $\langle s\rangle$. What exactly are these things? I have seen that bracket notation in Group theory. Mainly in relation ...
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19 views

Counting orbits and stabilisers

Let x denote the rotational symmetries of a cube. The vertex one can be taken to any other vertex by rotation so the orbit of 1 is $$orb_x (1)={1,2,3,4,5,6,7,8}$$ For the stabilisers I have $$stab_x ...
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21 views

What's the correct definition of generated ideal in a pseudo-ring?

Given a ring (with $1$) $R$, one defines what, say, a left ideal is. There's also a natural definition of ideal generated by a subset Definition A: $_R(S):=\bigcap\{I\supseteq S:I\text{ is a left ...
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Transcendence base of $\mathbb{C}$ over $\mathbb{Q}$ has infinitely many elements.

I need to show that transcendence base of $\mathbb{C}$ over $\mathbb{Q}$ has infinitely many elements. Since I do not know much about ordinals and cardinals, a proof based on algebra (rather than ...
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Rings where action of automorphisms on maximal ideals is transitive

If $R$ is a commutative ring, $\alpha: R \to R$ an automorphism of $R$, and $M$ a maximal ideal of $R$, then $\alpha(M)$ is also a maximal ideal of $R$ with the same quotient field. So the group of ...
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Product of disjoint cycle example

$(123)(45)(15)(24) = (14)(235)$, according to my lecture notes, yet I keep getting $(143)(25)$. By doing $$1 \to 4\\ 4 \to 3\\ 3 \to 1\\ 2 \to 5\\ 5 \to 2$$ Where am I going wrong?
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If $p$ is a prime, prove there are exactly $\frac{p^3-p}{3}$ monic irreducible cubic polynomials in $\mathbb{Z}_p[x]$

If $p$ is a prime, prove there are exactly $\frac{p^3-p}{3}$ monic irreducible cubic polynomials in $\mathbb{Z}_p[x]$ I am looking some notes here but don't know in general how to approach this ...
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is a number field by definition a subfield of $ \mathbb C $?

I have seen that some authors are defing the number field as a subfield of $ \mathbb C$ which is a finite extension of the rational numbers $ \mathbb Q $, while some others without referering to ...
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16 views

Distinction between algebra homomorphisms and $A$-module homomorphisms

I am getting quite confused about the distinction between algebra homomorphisms and $A$-module homomorphisms, where $A$ is an algebra. If $A=\mathbb CG$, the group algebra, then I have a result in my ...
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2answers
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behavior of a rational prime in quadratic extension (definition)

Let $ \mathbb Q \subset K=\mathbb Q (\sqrt{-n}) \subset L $, where $K/ \mathbb Q $ is a finite extension (i.e. $K$ is a number field) and $L/K$ is a maximal uramified abelian extension. If $p ...
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28 views

Be $K$ of characteristic zero and $f(x) \in K[x]$. Prove that $f'(x) = 0$ then $f(x)$ is a constant polynomial.

Be $K$ of characteristic zero and $f(x) \in K[x]$. Prove that $f'(x) = 0$ then $f(x)$ is a constant polynomial. I know that the field of zero characteristic is a field where any sum of multiplicative ...
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Let $V$ be the vector space over $R$ composed of all polynomials in **R[X]** having degree less than 3 [on hold]

can someone help me with this problem please. Let $V$ be the vector space over $R$ composed of all polynomials in R[X] having degree less than 3 and let $W$ be the vector space over $R$ composed of ...
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Showing $G(\mathbb{Q}(\sqrt[3]{2}, w)/ \mathbb{Q}) \simeq S_3$

I want to show that $G(\mathbb{Q}(\sqrt[3]{2}, w)/ \mathbb{Q}) \simeq S_3$, but I'm not that familiar with computing Galois groups so I don't really know how to do this exercise. How do I approach ...
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1answer
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Construction of a splitting field.

Building splitting field over $\mathbb Q$ of $f(x) = x^{5} - 3$. I know that the criterion of Eisenstein that $f(x)$ is irreducible with $p=3$. Then $f(x)$ has five distinct roots. I know that ...
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Decomposition of standard Borel subgroup.

I am reading the lecture notes. On page 3, formulas (1.7), (1.8), (1.9), let $P$ be the standard Borel subgroup, we have $P=MN$. Why $M, N$ must be (1.8), (1.9)? I know that $P$ should be a upper ...
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About annihilators in noetherian ring modules

Lately I've been thinking to annihilator of modules and I've conjectured a proposition I can't prove, so I'll expose my claim. Let $A$ be a noetherian reduced (commutative) ring and let $M$ be a ...
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Extend a homomorphism

Assume $B,H,G$ are abelian groups, $f:B\rightarrow H$ is a surjective homomorphism, $H$ is a subgroup of $G$. My question is :is there an abelian group $A$ and a surjective homomorphism ...
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Tell me whether this Unknown Operation Exists.

I need to know whether the below unknown operation, denoted by $\boxplus$ exists. If $v_1=a \boxplus X$ and $v_2=b \boxplus X$, where $X$ is an identical value in both $v_1$ and $v_2$: equation (1): ...
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Proving if $|G|=280$, then $G$ is not simple

So $280 = 2^3\cdot 5 \cdot 7$. I assume $G$ is simple. I'm having trouble using the Sylow Theorems to show that this is not Simple. In particular, computing the number of sylow groups and using that ...
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2answers
51 views

Definition of a prime

Whilst studying ring theory I came across the following definition of a prime: Given a ring $R$ we say that $r\in R \setminus \{{0}\}$ is prime if $r$ is not a unit, and $r|xy$ implies that $r|x$ or ...
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Cancellative Abelian Monoids II

Following this question I was asking myself if (in a cancellative Abelian Monoid $M$) given three elements $a,\, b,\, c$ for which there exists the least common multiple $m$, it will also exists the ...
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About the elements of Dihedral Group.

I have some difficulties finding the elements of Dihedral Group $D_8$. (Note that e.g., The order of $D_8 = 8$) I know The Geometric Approach for defining the $D_8$. But I always tend to like ...
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Elementary divisors theorem for Dedekind domains (Exercise in Lang's Algebra)

Exercise 13 (b) of Chapter III in Lang's Algebra is as follows. Let $M$ be a finitely generated projective module over the Dedekind ring $\mathfrak{o}$. Then there exists free modules $F$ and ...
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Central extension of the Discrete Heisenberg group $H_3(\Bbb Z)$

I want to use the Discrete Heisenberg group $(H_3(\Bbb Z),\times)$ as an example for a presentation on central extensions. $H_3(\Bbb Z) = \begin{bmatrix}1&x&z\\0&1&y\\0&0&1 ...
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How do I prove that the symmetric group $S_p$ where p is prime can be generated by any transposition and any p-cycle?

I am at a complete loss as to how to even begin. I think it has something to do with the fact that any p-cycle can be represented by $(123...p)$?
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Isomorphism proof between fields of polynomials with several indeterminates. [on hold]

Let $F$ be a field. Prove that the extension $F(x_1)(x_2) \cdots (x_n)$ is isomorphic to $F(x_1,x_2, \cdots, x_n)$ for indeterminates $x_1, x_2, \cdots , x_n$.
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Why does an isomorphism need to be a homomorphism?

In many books I read I found isomorphism defined as a 'bijective homomorphism'. I do not understand why is it that existence of inverse or order preserving requires the property of a homomorphism, ...
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Question about computing cohomology of trivial action on $\mathbb{Z}_{4}$

I'm currently considering the trivial action of the group $G = \mathbb{Z}_{2}$ on the group $A = \mathbb{Z}/4\mathbb{Z}$. It is easy to show that $|C^{2}(G,A)|$ = $2^{8}$ and that $|B^{2}(G,A)| \leq ...
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Product of characters in representation theory

Is it true that if $\phi_1$ and $\phi_2$ are characters then $\phi_1\phi_2$ is a character of a representation? I think this is true, say for instance if $R_1$ is a matrix representation with ...
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Please help me with proving a subgroup is normal

Question: Let $G$ be a group and $N = \{g_1h_1g_1^{-1}h_1^{-1} \dots g_nh_ng_n^{-1}h_n^{-1} \mid n \in \Bbb N, g_i, h_i \in G\}$. Show that $N$ is a normal subgroup of $G$. I have shown that $N$ ...
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Why $\ker N_{E/F}$ is a map from $E^{\times}$ to $F^{\times}$?

I am reading the lecture notes. In the end of page 1, it is said that $\ker N_{E/F}$ is a map from $E^{\times}$ to $F^{\times}$. Here $E/F$ is a quadratic field extension. Let $\alpha \in E$ and ...
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Why is the k-th convolution of $P_S$ is equal to ${P_S^k}$

For a random walk using transpositions on $S_n$, how can it be explained that the k-th convolution of $P_S$ is equal to ${P_S^k}$. They look to be the same intuitively but how can it written ...
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1answer
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What are the primary disadvantages of Dummit and Foote's abstract algebra text (3rd ed.)?

I have done a fair amount of research concerning which abstract algebra book to "settle down into"; that is, I wanted to pick an algebra text and really commit to it as my "primary text," more or ...
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2answers
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Ways to reduce polynomial modulus irreducible polynomial in F_2

Thank you all in advance for the help. I really appreciate your time in answering this question. I am trying to find a good summary of ways to reduce polynomial modules irreducible polynomial in ...