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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Show that it is a homomorphism?

For any abelian group $G$ we have $e_n: G \to G, e_n(g) = g^n$. By convention $e_0(g) = 1$. For a Field $F$ we have the subgroup $\{1,-1\} \leq F^*$. When $F$ is of characteristic $2$, this is the ...
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Prove that on N, the relation V where mV n

This is my question! Help me, please! Prove that on $\Bbb N$, the relation $\mathsf V$ is a linear order where $m\mathsf Vn$ if and only if $m$ is odd and $n$ is even, or $m$ and $n$ are even and ...
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Prove that the set A satisfies all the axioms to be a commutative ring with unity. Indicate the zero element, the unity and the negative.

A set $A$ with operation of addition and multiplication is given. Prove that the set $A$ satisfies all the axioms to be a commutative ring with unity. Indicate the zero element, the unity and ...
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40 views

What Notation is this?

When $p$ is prime, show that $v: Z^*_p \rightarrow U_2$ I know that the $Z_p$ is the elements $\{0,1,2,\cdots,p-1\}$ But what about the star on top of the $p$? Is that the group operation? Because ...
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4answers
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Finite dimensional algebra

Let $A$ be a finite dimensional algebra. Prove that an element of $A$ is invertible iff it is not a zero divisor. Let $a$ be an invertible element, then there exists an element $b$ such that $ab=1$ ...
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Subgroups and subsets

I have some trouble with groups. Say we know that A is a subgroup of B. If we have some subset of A, say H, can we deduce that H is also a subgroup of B? Thank you. So if I have set of 2x2 real ...
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17 views

Two questions concerning ideal factorization and norm

$\bullet$ In $\mathbb Z[\sqrt{-5}]$ why is $(2)=(2,1+\sqrt{-5})(2,1-\sqrt{-5})$ Actually both ideals on the RHS contain $(2)$, but also their product ? Can we just multiply RHS in the normal sense; ...
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Help to prove that a group is cyclic

As part of my study of Abstract Algebra I'm trying to prove that $U_p$ si cyclic for $p$ a prime number. It's a classical result, but I'm trying to prove it following 4 steps stated as problems in my ...
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2answers
46 views

Give an intuitive explanation for polynomial quotient ring, or polynomial ring mod kernel

I learned how to see quotient groups intuitively when I learned of a group mod its commutator subgroup. If we take a group and mod out all the elements that do not commute, we get a quotient group ...
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Polynomial and a field

How to prove that if a polynomial $$f(x) = ax^3+bx^2 +cx +d,$$ where $a,b,c,d \in K$, where $K$ is a subfield of $\mathbb{C}$, has a root in $K(\alpha)$ then $f$ has a root in $K$. $\alpha \in ...
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About Ito's theorem exemple

Find an exemple of Lie algebra $g$ that is metabelian and there are not two abelian algebras $A$ and $B$ such that $g=A+B$.
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Polynomial in $\mathbb{Z}_2[x]$ that is reducible but has no roots a prime $p$ for which $x+10$ divides $x^4+x^3+x+1$ in $\mathbb{Z}_p[x]$

First, I am suppose to find a prime $p\geq 4$ where $x+10$ divides $x^4+x^3+x+1$ in $\mathbb{Z}_p[x]$. Second, I am supposed to find a fifth degree polynomial in $\mathbb{Z}_2[x]$ that is reducible ...
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2answers
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Solvable Lie algebra

Let $g=A+B$ be a Lie algebra, where $A,B$ are metabelian algebras. Is $g$ solvable? By definition, an algebra $A$ is metabelian if $A\prime$ is abelian, where $A\prime=[A,A]$ .
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Integral domains examples

I am supposed to give an example of 1) an infinite integral domain of characteristic $5$, and 2) an integral domain which is not a field. Respectively, examples I chose were $\mathbb{Z}_5$ and ...
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1answer
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Algebra vs field extension. Examples of finite dimensional algebras without primitive generators.

Could someone clarify the difference, if any, between an algebra over a field $K$ and a field extension of $K$? Also, can someone provide an example of a finite dimensional algebra over $K$ that has ...
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Meaning of the term $X/H$ and orbits

I am trying to find representations of the group $G=GL_2(F(t)/t^2) = (M_2(F_p) , + ) \rtimes GL_n(F)$ So I was trying to do exactly what Serre has explained in this section. I am not quite able to ...
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Cyclic subgroups of $\mathbb{Z} /100\mathbb{Z} \oplus\mathbb{Z}/25\mathbb{Z}$

$\mathbb{Z} /100\mathbb{Z} \oplus\mathbb{Z}/25\mathbb{Z}$ has 24 elements of order 10. Why each cyclic subgroup of order 10 has four elements of order 10 ?
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Find the Number of Elements of a Particular Quotient Ring

Find the size of $\mathbb{Z}[\sqrt{-19}]/I$, where $I=(18+\sqrt{-19}, 7)$. The standard way to proceed would be $\mathbb{Z}[\sqrt{-19}]/I=\mathbb{Z}[x]/(x^2+19, 18+x, 7)=\mathbb{Z}_7[x]/(x^2+5, ...
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2answers
26 views

Group Theory and Lagrange's Theorem: coprime subgroups.

Let $G_1$ and $G_2$ be finite groups, and let $K≤G_1 \times G_2$. Let $H_1 = \{ g \in G_1 : (g,e) \in K\}$ and $H_2 = \{g \in G_2 : (e,g) \in K\}$ and suppose $|G_1|$ and $|G_2|$ are coprime. Then ...
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1answer
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How to find all roots of the equation $ x^3 + 2x^2 - 3x$ in $\mathbb Z_{12}$

Firstly you can factor it completely from $ x^3 + 2x^2 -3x$, which is $x(x-3)(x+1)$. We have the obvious roots of $0$, $3$ and $-1$, but what about the other roots? I have a little confusion here ...
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Can I represent groups geometrically?

I have just taken up abstract algebra for my college and my professor was giving me an introduction to groups, but since I like geometric definitions or ways of looking at stuff, I kept thinking, "How ...
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How to show $k$-algebras are isomorphic in practice

I am working through some problems which require me to show when some $k$-algebra ($k$ a field) maps are isomorphisms. Unfortunately, I've got myself a bit confused with definitions and the like, and ...
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Are ℚ/ℤ and ℚ isomorphic as (additive) groups? [on hold]

Is there an isomorphism $${\Bbb Q} / {\Bbb Z}\cong\Bbb Q$$ (of additive groups)? Justify your answer.
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Use of the Bezout's theorem in Abstract Algebra

The Bezout's theorem: Let $C$ and $D$ be two plane curves described by equations $f(X,Y) = 0$ and $g(X,Y) = 0$, where $f$ and $g$ are nonzero polynomials of degree $m$ and $n$, respectively. ...
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1answer
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Ideals and Null set

I was wondering if an Ideal in a ring can ever be the null set. The definition of an Ideal $I$ is that it is a subset of the ring $R$ such that: 1)It is an abelian group under "addition" (I put it in ...
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A free group is residually nilpotent

How can I prove that a free group is residually nilpotent group. Definition- A group G is residually nilpotent if for every non-trivial element $g$ there is a homomorphism $h$ from G to a nilpotent ...
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3answers
44 views

Prove: $e^x$ is transcendental over the polynomials with coefficients in $\mathbb{R}$

I have to prove the following for my math study: Prove: $e^x$ is transcendental over the polynomials with coefficients in $\mathbb{R}$. So far, I've done this: It's enough to prove that if ...
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1answer
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Show that this group is nilpotent.

Let $G$ be a finite solvable group whose order is divisible by at least three distinct primes. If every Hall $p'$-subgroup of $G$ is nilpotent, show that $G$ is nilpotent. I feel like the best ...
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Compatibility of direct product and quotient in group theory

This question came to me when I tried comparing direct product and quotients of groups with products and quotients of natural numbers. When we divide a number by another and multiply the result with ...
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Is $T_n(R) \cong T_n(R)^{op}$?

I am working on the following problem: Let $R$ be a commutative ring, and $T_n(R)$ be the ring of $n \times n$ upper triangular matrices. Is $T_n(R) \cong T_n(R)^{op}$? I have already shown ...
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centralizers of $X= (M_2(F_p),+)$ in $H=GL_2(F_p)$

I need to find the centralizers of $X= (M_2(F_p),+)$ in $H=GL_2(F_p)$ in order to find the action of $H$ on $X$ which will help me find the orbits of $X$ I Know that the centralizers of $M_2(F_p)$ ...
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Class number of $\mathbb Q(\sqrt{10}) $

I am interested in knowing how to compute the class number of $\mathbb Q(\sqrt{10}) $. I am confused with these class number computations.
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2answers
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Show that $B/Q$ is integral over $A/P$

If $A$ is a subring of $B$ and $B$ is integral over $A$, let $Q$ be a prime ideal of $B$ and $P=Q\cap A$. Show that $B/Q$ is integral over $A/P$. If $b\in B$ is integral over $A$ then for some ...
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Isomorphism and Quotient Ring [on hold]

Let $R$ be a ring. If for every proper ideal $I$ of $R$ we have $R/I\cong R$, then show that for every two proper ideals $I$ and $J$ of $R$ either $I\subseteq J$ or $J\subseteq I$.
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Show that $x^{8}+x^{4}+x^{3}+x+1$ is irreducible over $\mathbb{Z}_{2}[x]$

How do I show that $x^{8}+x^{4}+x^{3}+x+1$ is irreducible over $\mathbb{Z}_{2}[x]$? Someone says I should use the fact that the range of the matrix is 7, but I don't exactly know how that applies. ...
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29 views

ring of real functions field or not

Can somebody explain why $\cal{F}(\mathbb{R})$ is not a field nor an integral domain? On what instance does it not satisfy the definition of an integral domain?
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Intuition behind quotient groups?

I am having a hard time seeing the intuition behind quotient groups or rings. Intuitively, for a group, say Z/nZ would the quotient groups be the different sub groups of order 0 to n-1? Or how would ...
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What does it mean for something to hold “up to isomorphism”?

For example, to say that there are 2 such groups up to isomorphism such that the order of G is equal to $p^2$?
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1answer
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Finding roots in finite fields.

On pg. 587 (in the finite fields chapter) of Abstract Algebra, 3rd ed. by Dummit and Foote, the following statement is made: 'If $f_1(x)=x^4+x^3+1$, $f_2(x)=x^4+x+1$ are two of the irreducible ...
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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 ...
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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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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 $B\cong C$? What if I decompose B and C first, then ...
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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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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) ...
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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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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
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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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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 ...
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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
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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 ...