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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If $G$ is a field and there is an isomorphism $f\colon H/I \to G$, then does $I$ have to be a principal ideal?

I noticed that the ideal $I = \left(2, 1 + \sqrt{-7}\right)$ follows the definition of a non-principal ideal. I took two random elements from $\mathbb Z[-7]$, say $ a + b\sqrt{-7}$ and $ c + ...
2
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1answer
49 views

Are there any free or fascist boolean algebras?

A boolean algebra is an algebra with the binary operations $\wedge$, $\vee$, an unary operation $\neg$, and constants $0$, and $1$, satisfying axioms. A heyting algebra is an algebra with the binary ...
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0answers
10 views

A simple divisible module

Let $k$ be division ring and $V$ be a left $k$-vector space of infinite dimension. Let $E=End(V)$, defined as a ring of right operators on $V$. My questions are: (1) Is $V$ a simple right $E$-module? ...
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0answers
14 views

Indecomposable commutative rings

Let $R$ be a commutative ring. Can we say that $R=\bigoplus_{i\in I}R_i$ or $R=\prod_{i\in I}R_i$ where $R_i$ are commutative ring and $I$ is an infinite set?
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4answers
203 views

Why do we have “another” definition for the kernel?

Why does the definition $\ker(f)=\{(a,a')\in A\times A: f(a)=f(a')\}$ exist? This definition is for any sort of algebraic system and any sort of function. But which came first... this definition or ...
2
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1answer
47 views
5
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88 views

Is $(x)\otimes_{k[x]/(x^2)}(x)$ zero?

I am trying to decide if $(x)\otimes_{k[x]/(x^2)}(x)$ is zero. So I considered $x \otimes x$ which I rewrote as $1 \otimes x^2 = 1 \otimes 0 = 0$. But then I realized that $1$ does not live in ...
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0answers
21 views

Why is the coproduct $\Delta$ of the quantum double $D(H)$ an algebra homomorphism?

If $H$ is a a finite dimensional Hopf algebra, $H^\ast\otimes H$ has a Hopf algebra structure with multiplication $$ (\phi\otimes h)(\psi\otimes g)=\sum \psi_2\phi\otimes h_2g\langle ...
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2answers
39 views

Composition series and its number determine a group?

By Jordan-Holder thm, it is known that every finite group has a unique composition series.(Here, unique means that there is only one kinds of such series.) And it is known also that composition ...
3
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1answer
49 views

Frobenius automorphism and non-split Cartan subgroup of $\mathrm{GL}_2(\mathbb{F}_p)$

For completeness I give some definitions. Let $p$ a prime number and consider $V$ a 2-dimensional $\mathbb{F}_p$-vector space. Consider $k$ a sub-algebra of $\mathrm{End}(V)$ that is a field of $p^2$ ...
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4answers
110 views

Show that $t^n-1 \mid t^m-1 \Leftrightarrow n\mid m$

I want to prove the following lemma: $t^n-1$ divides $t^m-1$ in $F[t, t^{-1}]$ if and only if $n$ divides $m$ in $\mathbb{Z}$. I have done the following: $\Leftarrow $ : $n\mid m \Rightarrow ...
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1answer
40 views

Show that $(t^m-1)/(t^n-1)$ is a square if and only if $(\exists s \in \mathbb{Z})\ m=np^s$

I want to show the following lemma: Assume that the characteristic of the field $F$ is $p$ and $p>2$. Then $(t^m-1)/(t^n-1)$ is a square in $F[t, t^{-1}]$ ($F[t,t^{-1}]$: the polynomials in ...
4
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2answers
57 views

Failure of group definition with weaker axioms

In "A First Course in Abstract Algebra", Edition 7, p.43, Fraleigh writes that It is possible to give axioms for a group $\left<G,*\right>$ that seem at first glance to be weaker, namely: ...
6
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2answers
446 views

Is ideal an “anti-field”?

I am comparing theorems on normal subgroup and ideal from Fraleigh's, and come to this strange intuition. I hope my conclusion does not screw up, I hope I won't get ridiculed: Theorem 15.18: $M$is ...
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0answers
40 views

Polynomial Interpolation and Data Integrity

This question is about polynomial interpolation and security. Please consider a scenario where we have a polynomial $f$, one of whose roots is $a$. We evaluate it at some ...
4
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1answer
29 views

Ideals of non semi-simple group rings.

I worked for a long time on complex group rings and complex twisted group rings. In those cases the algebra is semi-simple and its structure is well understood from the decomposition to irreducible ...
4
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1answer
70 views

If $m$ divides $n$, find a free resolution of $\mathbb{Z}/m$ as a $\mathbb{Z}/n$-module.

If $m$ divides $n$, find a free resolution of $\mathbb{Z}/m$ as a $\mathbb{Z}/n$-module. I have tried this one and got $0 \leftarrow \mathbb{Z}/m \leftarrow \mathbb{Z}/n \leftarrow \mathbb{Z}/n$. ...
9
votes
6answers
845 views

Dependence of Axioms of Equivalence Relation?

This question is problem 11(a) in chapter 1 in 'Topics in Algebra' by I.N. Herstein. These are the properties of equivalence relation given in this book. Prop 1 $a \sim a$ Prop 2 $a \sim b$ ...
3
votes
3answers
271 views

What is the “grade” in geometric algebra

I'm reading a book (Linear and Geometric Algebra, by Alan Macdonald) where the author uses the term grade without ever defining it. I have a murky sense of what the grade of a blade may be (a ...
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0answers
64 views

Sets with $n$ prime numbers

Let $n>2$ a natural number. We define the following sets: $$S=\{1 \leq a \leq n : (a,n)=1, a^{n-1} \not\equiv 1\pmod n\} \\ T=\{1 \leq b \leq n : (b,n)=1, b^{n-1} \equiv 1 \pmod n\}$$ ...
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2answers
41 views

Question about a passage in Fulton and Harris

So I was reading the first chapter of Fulton and Harris and they are determining the representations of $S_3$. I came along this passage and had some questions What do they mean when they say "the ...
3
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2answers
56 views

Kernel of ring homomorphism

Let $\phi: R \to R'$ be a ring isomorphism and $I$ an ideal of $R$. Define $\phi(I)=\{\phi(i): i \in I\}$. Show that $\frac RI \cong \frac {R'}{\phi(I)}$. To use the first isomorphism theorem, ...
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2answers
139 views

compatible operations on the integers?

how does one describe all partitions on the integers that are compatible with the operation +? By compatible it is means that for any 2 sets in the partition $A,B\in P$, and every combination of ...
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1answer
62 views

Factoring $x^{15}−1$ into irreducible polynomials over $\mathrm{GF}(2)$

Factorize $x^{15}−1$ into irreducible polynomials over $\mathrm{GF}(2)$ The answer is $$(x+1)(x^2+x+1)(x^4+x+1)(x^4+x^3+1)(x^4+x^3+x^2+x+1)$$ but how would I find this? Please help.
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1answer
29 views

Difference between torsion and zero divisor

I'm not understanding what the difference is between a zero divisor and a torsion element of a module. My best guess is that the torsion elements are "vectors" and zero divisors are scalars. This ...
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0answers
36 views

Inducing representations from a subgroup of finite index.

Let $G$ be a group and $H$ a subgroup of finite index. Let $(\sigma , W) $ be a irreducible representation of $H$ (which need not be finite dimensional). 1) When can we extend this representation of ...
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1answer
26 views

What is the name for this $R$-module? [on hold]

If $M$ is a $R$-module such that for all $x,y$ in $R$ and $m$ in $M$ then $x.y.m=0$ ... then: how is called $M$?
2
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1answer
47 views

Problem involving cubic field extensions

Let $F$ be a field of characteristic $0$ and let $L$ be a cubic extension. I want to show that there exists an element $a \in F,$ and an extension $L_0$ of $\mathbb{Q}(a)$ such that ...
4
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0answers
36 views

Splitting even degree polynomials

I have an octic equation (degree $8$) and a sextic equation (degree $6$) in $\Bbb Z[x]$ with very large coefficients (size several hundred bits) that I know splits into two quartics and two cubics ...
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2answers
33 views

Ring homomorphism and ideal that contains the kernel [on hold]

If $f:R\rightarrow S$ is a ring homomorphism and $I$ ia an ideal of $R$ such that $ker(f) \subseteq I$ then $f^{-1}(f(I))=I$ We know that $I\subseteq f^{-1}(f(I))$ but how can I use that $ker(f) ...
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1answer
15 views

Recurrence Relations for Sequence Counting Hamming Weights

Define $a(n,k)=|\{x \in \mathbb{F}^{2n}_2 : Hx=0, ||x||=k\}|$ and $b(n,k)=|\{x \in \mathbb{F}^{2n}_2 : Hx=1, ||x||=k\}|$ where $||\cdot||$ denotes the Hamming weight of $x$ (i.e. number of non-zero ...
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1answer
39 views

A symmetric group question. [on hold]

Determine the integers $n$ such that there is a Surjective homomorphism from the symmetric group $S_n$ to $S_{n-1}$. It is a question from Artin's book. Exercise 7.5.8
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1answer
97 views

Solve $x^3+x+3=0$ by Galois's theory

Solve with radicals the following equation $x^3+x+3=0$, using Galois Theory. I'm just starting learning this and I do not have many ideas.
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1answer
234 views

“Wheel Theory”, Extended Reals, Limits, and “Nullity”: Can DNE limits be made to equal the element “$0/0$”?

"Wheels" are a little-known kind of algebraic structure: They modify the concept of a field or a ring in such a way that division by any element is possible, including division by zero, while also ...
2
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2answers
105 views

integral ring extension, maximal ideals

Let $\varphi:A\rightarrow A'$ be an integral ring extension. 1) Show that for every maixmal ideal $m'\subset A'$ the ideal $\varphi^{-1}(m')\subset A$ is maximal 2) and that for every maximal ...
2
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1answer
49 views

If $\Delta$ is cocommutative in $k(G)$, why does that imply $G$ is abelian?

Suppose $G$ is a finite group and $k(G)$ is it's group function Hopf algebra. I read that for $k(G)$ is quasi-triangulated requires that $G$ be abelian. If $R$ is the distinguished element of ...
2
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1answer
27 views

Why does $\operatorname{Ad}_h((S\otimes 1)(Q))=\epsilon(h)(S\otimes 1)(Q)$ in a quasi-triangular Hopf algebra?

I'm reading a proof that in a quasi-triangular Hopf algebra $H$, $(S\otimes 1)Q$ is $\operatorname{Ad}$-invariant. Here $Q=\tau(R)R$, where $R$ is the invertible element in $H\otimes H$ satisfying all ...
1
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1answer
43 views

For each $n >2$ find all $m \in \mathbb N$ such that there exist a group $G$ of order $m$ and a surjective homomorphism $f: A_n \rightarrow G$

For each $n >2$ find all $m \in \mathbb N$ such that there exist a group $G$ of order $m$ and a surjective homomorphism $f: A_n \rightarrow G$< I tried to use the theorem of isomorphism, and a ...
2
votes
1answer
34 views

Showing a nonabelian group of order 21 has an automorphism that is not inner.

I've seen at least 3 proofs of this on here, but most involved techniques I don't think I'm comfortable with, so I wanted to see if this one works: Since $21=3\cdot 7$, up to isomorphism there's only ...
4
votes
1answer
35 views

A $\mathbb Z/p\mathbb Z[G]$ submodule with no complement

Let $G$ be a group acting on a set $X$ of size $n$. Suppose $G$ acts doubly transitively. If $p$ is a prime, this naturally gives a permutation representation on the vector space over $\mathbb ...
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0answers
39 views

A problem about isomorphism in module theory

For a sequence of $R$-modules like this:$A \overset{f}{\rightarrow} B \overset{g}{\rightarrow}C$ such that $\mathrm{Im} f \subset\ker g$ and $B/\mathrm{Im}f \cong B/\ker g$. Then $\mathrm{Im}f=\ker ...
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1answer
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Is the ring of holomorphic functions on $S^1$ Noetherian?

Let $S^1={\{ z \in \Bbb{C} : |z|=1 \}}$ be the unit circle. Let $R= \mathcal{H}(S^1)$ be the ring of holomorphic functions on $S^1$, i.e. the ring of functions $f: S^1 \longrightarrow \Bbb{C}$ which ...
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1answer
44 views

Prove that a morphism $\alpha$ of $Fun(\mathcal{A},\mathcal{B})$ is an isomorphism iff each component $\alpha_A$, is an isomorphism in $\mathcal{B}$

I'm a computational engineer starting with a course of Introduction to Category Theory, and perhaps is extremely basic what I'm asking but I'm trying to learn how to make proofs in category theory ...
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2answers
53 views

If $G = S_5$ and $H = \{g \in G \mid g^{5} = e\}$ how could I determine and prove whether or not $H$ is a subgroup of $G$?

I think that the this group contains the 5 element cycles and the identity e but overall I'm not sure how to prove that the product of the 2 members of H is also a 5 cycle or e.
3
votes
3answers
68 views

Why does $\sqrt{6} + \sqrt{10} + \sqrt{15}$ have four conjugates?

I am having trouble understanding how algebraic number $\sqrt{6} + \sqrt{10} + \sqrt{15}$ has four conjugates. Minimal polynomial is $x^4-62 x^2-240 x-239$ according to Wolfram Alpha. Factorized: ...
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0answers
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Groups,subgroups, normal subgroups [on hold]

Let $G$ be the group of all $2\times 2$ real matrices $\left( \begin{array}{ccc} a & b \\ 0 & d \end{array} \right) $ under matrix multiplication where $a,d\neq 0$. If $N=\left\{ \left( ...
3
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1answer
42 views

What do we know about fields possessing an involution?

The field $\mathbb{C}$ of complex numbers has an involution, and the same is true of the field of algebraic numbers (the algebraic closure of $\mathbb{Q}$ as a subfield of $\mathbb{C}$) and of the ...
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2answers
2k views

Why are higher-degree polynomial equations more difficult to solve?

I am confused about the significance of the powers on equations. For example, in $ax = b$, intuitively $b$ is a value $x$ multiplied $a$ times. In $ax + b = c$, $c$ is a value $x$ multiplied $a$ times ...
0
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1answer
31 views

Properties of isomorphism in module theory [on hold]

I have two exercises, but I can't solve them: a. If $X, A, B $ are $R$-modules with $A \subset B \subset X $, prove that if $X/A \cong X/B$ then $A=B$ b. If $X/A \cong X$ then $A=0$
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4answers
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The transpose of a permutation matrix is its inverse.

This is a question from the free Harvard online abstract algebra lectures. I'm posting my solutions here to get some feedback on them. For a fuller explanation, see this post. This problem is from ...