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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There exists a homomorphism $f : G \to H$ with $|G| = 20$ and $|im f | = 6$

There exists a homomorphism $f : G \to H$ with $|G| = 20$ and $|im f | = 6$? Is this true? I know that I have to use the first isomorphism theorem but I don't know what to do next?
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Quotient of graded rings by graded ideals are again graded rings

I want to prove the following statement. Let $A = \bigoplus_{n=0}^{\infty}A_n$ to be a graded $R$-algebra and $I$ a graded ideal of $A$. Let $(A/I)_n = (A_n + I)/I$ be the image of $A_n$ in $A/I$. ...
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Boolean algebras and rings

I know that M. H. Stone proved that there is a bijection between boolean algebras and boolean rings. The bijection I know is the following: to any given Boolen algebra $(L,\, \vee, \wedge)$ we ...
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R has IBN but R fails rank condition

I need an example about "IBN for ring": R is a ring (no commutative), R has IBN but R fails rank condition. thanks
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20 views

Tensor algebra becomes an R-algebra. Theorem, Proof, Dummit and Foote

I have the definition of tensor algebra as follows: $T(M) = \bigoplus_{i =1}^{\infty} T^i(M)$, where $M$ is an $R$ module, where $R$ is commutative and contains the element $1$. Finally $T^k(M) = ...
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In any integral domain, only $1$ and $-1$ are their own multiplicative inverses.

In any integral domain, only $1$ and $-1$ are their own multiplicative inverses. Note that $x=x^{-1}$ iff $x^2=1$ I'm not sure how to go about proving this. I know the definition of an ...
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1answer
23 views

Image and kernel of the natural projection from a group to its quotient by a normal subgroup [on hold]

Let $H$ be a normal subgroup of a group $G$. Prove that the function $$f : G \to G/H$$ defined by $f (g)=gH$ is a homomorphism with image $G/H$ and kernel $H$. Use this fact to conclude that a ...
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1answer
24 views

Why is F($\beta$) a subfield of F($\alpha$)?

There is a corollary in my book that says: If E is an extension field of F, $\alpha \in E$ is algebraic over $F$, and $\beta \in F(\alpha)$, then $\deg(\beta,F)$ divides $\deg(\alpha,F)$. In ...
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1answer
39 views

How can one find irreducible elements in $\mathbb{Z}[\sqrt{2}]$?

Is it just all elements that have a prime norm? Ok, so from the comment I've seen that elements with a non prime norm can be irreducible...what exactly do I have to do to find irreducibility?
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Is this generalization of Boolean algebras a variety?

I'm interested in the class of structures $\langle S,\top,\neg,\wedge\rangle$ defined by the axioms: $p \wedge q=q \wedge p$ $p \wedge (q \wedge r) = (p\wedge q)\wedge r$ $p = p \wedge p$ $p = ...
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1answer
11 views

Ring polynomial kernel generators

This is the textbook question: Q: Find generators for the kernels of the following maps: $\mathbb{R}[x,y] \to \mathbb{R}$ defined by $f(x,y) \mapsto f(0,0)$ $\mathbb{R}[x] \to \mathbb{C}$ defined ...
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1answer
48 views

Can one define alternative algebras? [on hold]

Context I am an engineer by profession, and have read up on math some - where I got the sense that it was common to define alternative algebras for specific domains - where you define the operators, ...
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0answers
36 views

Sylow p-subgroups and set X not divisible by p

Let $P$ be a Sylow $p$-subgroup of $G$ and suppose that $P\subseteq Z(G)$. Show that the set $X$ of elements of $G$ with order not divisible by $p$ is a subgroup of $G$ and that $G=P\times X$. I ...
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2answers
33 views

Isomorphism of Localization

I believe, though a not sure, that any two ideals $A, B$ of a dedekind domain $X$ are isomorphic as $X$-Modules iff their localizations $A_p, B_p$ are isomorphic for any prime ideal $p$. Could anyone ...
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2answers
30 views

prove the nomalizer $N(H)$ of the subgroup $H$ in $G$ is a group

I need some help on the following question. For an arbitrary subgroup $H$ of the group $G$, the normalizer of $H$ in $G$ is the set $N(H) = \{x \in G \mid xHx^{-1} = H\}.$ Any help??
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If $P$ is a prime then $R/P$ is an integral domain.

I know the same question has been already asked here. So, I am not asking for any proof rather to find out what's wrong with my proof. So, this is what I did: Let, $a+p, b+p \in R/P$, since $P$ is a ...
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Quotient Groups with symmetric $S_4$ group [duplicate]

I'm working on this problem and I am having trouble figuring it out. The problem is: Find the quotient group $G/H$. Write out the distinct elements of $G/H$ and construct a multiplication table of ...
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Let $F,K$ be two fields $F \subset K$ and suppose $f(x),g(x) \in F[x]$ are relatively prime in $F[x].$ Prove they are relatively prime in $K[x].$

Let $F,K$ be two fields $F \subset K$ and suppose $f(x),g(x) \in F[x]$ are relatively prime in $F[x].$ Prove they are relatively prime in $K[x].$ Suppose $f(x)$ and $g(x)$ are relatively prime in ...
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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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1answer
21 views

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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2answers
57 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
255 views

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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3answers
28 views

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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18 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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2answers
35 views

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
53 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 [on hold]

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 [on hold]

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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1answer
15 views

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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1answer
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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
21 views

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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0answers
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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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1answer
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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
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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
26 views

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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1answer
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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
21 views

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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30 views

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
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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
38 views

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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1answer
20 views

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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26 views

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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38 views

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.