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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Are both of these proofs of Cayley's Theorem by Group Action?

The Wikipedia page, gave a proof of Cayley's Theorem, and then gave an alternative one that uses the language of group action. However, doesn't the first proof also uses a group action, which is the ...
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651 views

Who named “Quotient groups”?

Who decided to call quotient groups quotient groups, and why did they choose that name? A lot of identities such as $$\frac{G/A}{B/A}\cong \frac{G}{B}$$ suggest that whoever invented the notation ...
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1answer
46 views

Transcendence degree of fraction field

Let $k$ be a field and $p \in k[x_1, \dots, x_n]$ an irreducible element. Is there an elementary way to prove that $\operatorname{tr.deg}_k \mbox{Frac}(k[x_1, \dots, x_n]/(p)) = n-1$?
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1answer
37 views

Show that $\lambda_1 = \min \{ Q(u) \mid \|u\| = 1 \}$ and $\lambda_m = \max \{ Q(u) \mid \|u\| = 1 \}$

Let $V$ a vector space over $K$ and $Q(u) = \langle u, Tu \rangle$ a quadratic form. $T$ is a symmetric operator. The eigenvalues of $T$ are sorted by size $\lambda_1 < \dots < \lambda_m$. How ...
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1answer
41 views

Binomial Coefficients form Basis for Rational Polynomials

How would we show that the polynomials $c_n(x):=\dbinom{x}{n}$ form a basis for $\mathbb{Q}[x]$?
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4answers
60 views

About how to understand the third isomorphism theorem

What is a good way to understand the intuition behind the third isomorphism theorem? Is it something looking like zooming out of the group structure?(i.e. discard the detailed info by modding out a ...
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2answers
277 views

Quotient ring isomorphism

I think that if $A$ is any commutative ring with unity and $q\in A$, $p\in A[x]$ then we have $A[x]/(q,p)\cong A/(q)[x]/(\bar{p})$ where $\bar{p}$ denotes the class of $p$ in $A/(q)[x]$. Is this true? ...
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20 views

In an infinite cyclic field of non zero units, characteristic $\neq 2$, can an element $-u \neq u$ be expressed as $u^t$ for some finite integer $t$?

For the sake of a proof using contradiction ( to be used somewhere), Lets assume that an infinite cyclic field $F$ of non zero units exists with characteristic $\neq 2$ . In this infinite cyclic field ...
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2answers
22 views

Proof: Every normal subgroup has corresponding Congruence relation and vice versa

I am trying to prove the claim in the title. I was able to do most of the work, but I still need some help. I will show what I have written so far, and will highlight the parts in the proof that I ...
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1answer
30 views

Involution on semigroups with identity

I'm trying to understand the following: Let $S$ be a semigroup. By an involution on $S$ we mean a map $* : S \to S$ satisfying for all $a,b\in S$ $(ab)^*=b^*a^*$ $(a^*)^*=a $ My problem is the ...
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58 views

Existence of a coproduct and representability of a functor

I've found this claim: Let $\mathcal{C}$ be a category; the family $\lbrace C_i \rbrace_{i \in I }$ has a coproduct in $\mathcal{C}$ if and only if the functor $$F : \mathcal{C} \to Set$$ $$A ...
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18 views

Trace functionals as invariant elements of $R[\mathfrak{g}]$ under $G$

Let $\mathfrak{g}$ be a semisimple Lie algebra over $\mathbb{C}$ and let $G$ be its inner automorphism group. Then $G$ acts on $R[\mathfrak{g}]\cong S(\mathfrak{g}^*)$ via $(\sigma\cdot f)(x) = ...
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1answer
35 views

Free objects are generated by the image of the canonical injection

Let $\mathcal{C}$ be a concrete category and $X$ be a set, let $F_X$ be free on $X$ with canonical injection $i:X\to F_X$. Is it always true that $i(X)$ generates $F_X$? It looks like an easy ...
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1answer
45 views

What is the difference between $\mathcal{X}\subseteq\mathbb{R}^n$ and $\mathcal{X}\subset\mathbb{R}^n$

Let $\mathcal{X}$ be a non-empty set. For instance, let it be a set of vectors of the form $\mathbf{x}\in\mathbb{R}^n$, i.e., ...
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1answer
59 views

Given ring and ideal, How to prove that the intersection of ideals is an ideal

Given $R$ is a ring, $X\subseteq H_i$ and $H_i$ is an ideal of $G$ for each $i=1,2,...,n$. Prove that $H_1∩H_2∩...∩H_n$ is an ideal of $G$ and contain $X$. That is a question I get from random ...
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1answer
35 views

Axioms of associative algebra?

I am struggling to find a definition of an associative algebra. Wikipedia (http://en.wikipedia.org/wiki/Associative_algebra) says Let $R$ be a fixed commutative ring. An associative ''R''-algebra is ...
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2answers
40 views

For which $n,k > 1$ does $n\mathbb{Z}_k$ have a multiplicative identity?

By "ring," let us mean a not-necessarily unital ring. Then $2\mathbb{Z}_6 = \{0,2,4\} \subseteq \mathbb{Z}_6$ and $2\mathbb{Z}_8 = \{0,2,4,6\} \subseteq \mathbb{Z}_8$ are both rings under the induced ...
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1answer
33 views

Suppose that $F$ is a field of characteristic $\neq 2$ and non zero elements of $F$ form a cyclic group under multiplication. Prove that $F$ is finite

$1.$ Suppose that $F$ is a field of characteristic $\neq 2$ and that the non zero elements of $F$ form a cyclic group under multiplication. Prove that $F$ is finite Attempt : Let ...
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26 views

Why is the subfield (of a field) generated by an algebraic element equal to the subring generated by the same element?

I am trying to prove that, for a field extension $\mathbf{K}/k$ and $a$ an algebraic element over $k$, $$k(a)=k[a],$$ where $k(a)$ is the subfield of $\mathbf{K}$ generated by $a$ and $k[a]$ is the ...
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1answer
39 views

A field extension of degree 8

I would really appreciate it if you give me a hint on the following question: If $K \subset F$ is a field extension of degree 8, then we must have $F=K(a,b,c)$ for some a, b and c in F.
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2answers
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Intersection of an arbitrary subgroup with one of finite index

I want to show that if $G$ is a group and $[G:H]$ is finite, then so is $[K:H \cap K]$ for any $K < G$. I think I can do this by showing that $k \in K \implies k(H \cap K) = (kH) \cap K$. Is ...
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1answer
97 views

Additive non-abelian group?

Sometimes I see in books the term "additive abelian groups". In my opinion, when we use addition to represent the group operation, we already have in mind that the operation is commutative. So ...
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1answer
71 views

(True/False)? If $R$ is a commutative ring, then prove that $(s \cdot t) x =(s \cdot x)(t \cdot x) ; x \in R~,~s,t \in Z$

If $R$ is a commutative ring, then prove that $(s \cdot t) x =(s \cdot x)(t \cdot x); x \in R ~,~ s,t \in Z$ Attempt: I do not understand how $(s \cdot t) x =(s \cdot x)(t \cdot x) = (s \cdot ...
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1answer
78 views

Let $a_1,a_2,a_3,…,a_n$ be elements of a group $(G,*)$. Show by induction that $a_1*a_2*a_3…*a_n$ always gives the same answer.

Basically I need to prove that a group $(G,*)$ is associative in the general case. To do this I know I have to use induction to show that no matter where I insert parentheses into the equation ...
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2answers
31 views

Presheaves over a discrete space are necessarily sheaves?

In problem number 5.42 on p.302 of Homological Algebra text by Rotman it is asked to prove that every presheaf of abelian groups over a discrete space is a sheaf. However it looks to me that I have a ...
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1answer
97 views

Find a closed subset of an algebraic group, closed under products, which does not contain $e$.

The accepted answer for this question proves the following statement: If $S$ is a closed subset of an algebraic group $G$ which contains $e$ and is closed under taking products in $G$, then $S$ is ...
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1answer
36 views

Determining a linear relation via the wedge product.

The Grassmann algebra provides for an alternative of generalizing the quaternion algebra as opposed to the Clifford algebra. Here we define the wedge product $u \wedge v $. Now there exists a ...
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Is the completion of a Dedekind domain a PID?

Ths is a basic question on Dedekind domains. Let $R$ be a Dedekind domain, $P$ a non-zero prime ideal of $R$. I know that the localization $R_P$ is a PID, but is it true that the completion ...
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3answers
64 views

Cosets/ Cyclic group

Let $G=\langle a\rangle$ and $H=\langle a^2\rangle$. Find all the right cosets of $H$ in $G$. Additional info: I understand that a right coset of $H$ in $G$ is of the form $Ha=\{ha:h \in H\}$. But I ...
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2answers
57 views

Role of Commutative Ring? Suppose that $R$ is a commutative ring with no zero divisors, then Char $R$ is $0$ or prime.

There's a question in Gallian: Suppose that $R$ is a commutative ring with no zero divisors. Characteristic of $R$ is $0$ or prime. I am wondering what could be the role of $R$ given as being ...
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1answer
25 views

the greatest common divisor of three homogeneous polynomial in three variables

Let $f_1(x,y,z), f_2(x,y,z), f_3(x,y,z)\in k[x,y,z]$ three homogenous polynomials of the degree $d_1,d_2,d_3$, with $ k$ is algebraically closed. I need to show what the conditions of the ...
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104 views

Weak Amalgamation Property for Boolean algebras

I'm trying to study universal algebra and lattice theory by myself. Just got stuck with an exercise from Gratzer's "General Lattice Theory" and it seems to me that I don't fully understand the notion ...
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1answer
23 views

Frobenius theorem on real division algebras

Can someone help me with this? I don't understand conclusion after Exercise 7, that A is direct sum of eigenspaces U(1) and U(-1). Thanks in advance :) ...
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1answer
39 views

Exists a binary primitive pentanomial of degree $n$, for every $n\ge5$?

Or, in other words, prove (or disprove) this conjecture: $\forall n\ge5,\exists(i,j,k),n>i>j>k>0,\text{ such that}$ $\;x^n+x^i+x^j+x^k+1\text{ is a primitive polynomial in }GF(2)$. Also: ...
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0answers
48 views

When is an $R$-projective module a projective module?

Let $R$ be a semiperfect ring. Is it a true fact that every $R$-projective module $M$ with $Rad(M)$ superfluous in $M$ is projective? I could not reach a good result using just the fact that ...
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2answers
81 views

Dimension of $\Bbb Q(e)$ over $\Bbb Q$?

The dimension of $\Bbb Q(\sqrt{2})$ over $\Bbb Q$ is finite since $\sqrt2$ is algebraic over $\Bbb Q$. But what about any transcendental number (say $e$)? Which is the smallest field containing $\Bbb ...
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1answer
55 views

Let $(G,*)$ be a group and $g$ be a fixed element of $G$. Prove that $G=\{g*x \mid x \in G\}$

Let $(G,*)$ be a group and $g$ be a fixed element of $G$. Prove that $G=\{g*x \mid x \in G\}$ My proof: Let $(G,*)$ be a group and $g$ be a fixed element of $G$. Let $x=g^{-1}*y$, where $y$ is ...
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1answer
203 views

Isomorphism from $B[y]/IB[y]$ onto $(B/I)[y]$

For some reason I can't crack the following problem: Let $B$ be a ring, $I$ an ideal, and $A := B[y]$ the polynomial ring. Construct an isomorphism from $A/IA$ onto $(B/I)[y]$. How to ...
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How similarity transformation is related to coordinate transformation?

I know that every matrix can be transformed into its Jordan form using similarity transformation. But I wanted to know, this transformation is related to shifting of coordinate systems?
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1answer
66 views

Ring Structures On $\mathbb {R} ^n$

In the book of Musili it is written that $\mathbb{R}^n$ is a division ring under usual addition and multiplication for $n=1,2,4$. I have understood this. But after that he said, in those cases we ...
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1answer
24 views

Field Isomorphisms between a field and something that contains it

Are there any k-isomorphism of fields between M and L such that K $\subseteq$ M $\subset$ L? Examples would be appreciated. Thanks
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1answer
27 views

A question on index of subgroup in Dummit and Foote 3.2 Ex 10.

Suppose $H$ and $K$ are subgroups of finite index of a group $G$ (which maybe infinite), $|G:H|=m$ and $|G:K|=n$ prove that $\operatorname{l.c.m.}(m,n)\leq|G:H\cap K|\leq mn$. I am not sure about my ...
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1answer
53 views

The reals as an algebra over the rationals

R, the real numbers, is an infinite dimensional commutative division algebra over the rationals Q. Is there an example of an infinite dimensional noncommutative division algebra over the rationals Q?
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2answers
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Suppose that $R$ is a commutative ring with no zero divisors. Show that all non zero elements of $R$ have the same additive order.

Suppose that $R$ is a commutative ring with no zero divisors. Show that all non zero elements of $R$ have the same additive order. Attempt: CASE $1$ : When $R$ is finite commutative Ring Every ...
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What is Abstract Algebra essentially?

In the most basic sense, what is abstract algebra about? Wolfram Mathworld has the following definition: "Abstract algebra is the set of advanced topics of algebra that deal with abstract algebraic ...
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1answer
30 views

Surjective Homomorphism $D_{12}$

I'm trying to find all groups $H$ up to isomorphism such that there is a surjective homomorphism from $D_{12}$ onto H. The possible $H$ are the factor groups $D_{12}/N$ where $N$ is normal in $G$. ...
3
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1answer
47 views

Kernels of power surjective maps

Suppose $k$ is an algebraically closed field, and $A$ and $B$ are finitely generated, commutative, graded $k$-algebras. Suppose $\varphi:A\to B$ is a map of $k$-algebras. Notice if $B$ is a domain, ...
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1answer
61 views

Triviality of $H_3(G,\mathbb{Z})$

We know that the triviality of the Schur multiplier means that projective representations can be lifted to ordinary ones. The Schur multiplier is also a measure of the failure of how the commutator ...
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1answer
29 views

Proof that $o(\sigma)=3$ $\Rightarrow$ $\sigma \in$ Conjugacy Class of $(1 2 4)$

How can I show that $o(\sigma)=3$ $\Rightarrow$ $\sigma \in$ C($(1 2 4))$ in $S_4$? All I have to go with is that $\sigma ^3 = I$. I considered the use of Cayley's theorem, which would mean that a ...
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1answer
56 views

Generating a subgroup

While getting ready for my exams I am trying to solve this question "Find a Sylow $2$-subgroup in $S_4$ which contains the element $(1, 3, 2, 4)$." I started by permutation of the element required ...