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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Binary operation on Graphs

Are there "binary operations" on graphs, which make the set of all graphs, a commutative ring or a field For example $G_1 \cdot ( G_2 + G_3) = G_1 \cdot G_2 + G_1 \cdot G_3$ By a graph I mean ...
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36 views

For each polynomial $p \in K[t]$, there exists another polynomial $g$ such that $\{p(\epsilon_i): 1\leq i \leq n\}$ is its set of roots

I'd like to solve the following problem: Let $K$ be a field, $f \in K[t]$ a polynomial of degree $n$ and $E$ a splitting field of $f$ over $K$ in which $f$ has $n$ distinct roots ...
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2answers
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The product $J^{n}A$ of ideals

Suppose that $A$ and $J$ are two ideals of a ring $R$. I can't understand the following implication: If $JA = A$ then $J^{n}A = A$ for all $n > 0$. True that $J^{n}A$ is an ideal of $A$ for all ...
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1answer
51 views

Prove there isn't a non-zero nilpotent element of a finite ring

Let $(A, +, \cdot)$ be a finite ring with $n$ elements having the property: $x^n \ne1, \forall x \ne 1, x \in A$. Prove $0$ is the only nilpotent element of $A$. My attempt Suppose there is ...
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problem about inner semidirect product

Let $G$ be a group which is the product $G=NH$ of subgroup $N,H\subset G$ where $N$ is normal. Let $N\cap H=\{1\}$. I am trying to show that that there is an iso $G\cong N\rtimes H$, with the ...
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16 views

Grobner basis and number theory [closed]

Could someone help me? What is the relationship between grobner basis and number theory?
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1answer
75 views

In $A$-Mod, $M\oplus A\cong A\oplus A$ implies $M\cong A$

(Exercise from an introductory course in homological algebra) Whenever $A$ is a commutative ring with unit and $M$ an $A$-module, the following holds: $$M\oplus A\cong A\oplus A \Rightarrow ...
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To show that Z(G) = $\cap_{a \in G} C(a)$

To show that Z(G) = $\cap_{a \in G} C(a)$ (Intersection of all subgroups of form C(a)) Let $a \in Z(G)$. Then $ax=xa$ for all $x$ in G. IN particular we can say that $ax_1=x_1a$ and $ax_2=x_2a$ and ...
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1answer
50 views

Problem about subgroup of $D_n$

Prove that every subgroup of $D_n$ , either every member of subgroup is a rotation or exactly half of them are rotations. Intuitively, if every member is a rotation then they will form a subgroup ...
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34 views

When will the classification of a property of a group be complete

I am studying pronormal Hall $\pi$-subgroups of a finite group $G$ All Hall $\pi$-subgroups of a finite solvable groups are known to be pronormal as this follows from Hall's theorem Recently, it has ...
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32 views

How to prove isomorphic? [duplicate]

(Q*,•) and (R*,•) is isomorphic. This is false. Why this problem false?? Would you ask to me counter-example?
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23 views

Suppose n is even positive integer and $H$ is a subgroup of $\mathbb Z_n$ [duplicate]

Suppose $n$ is even positive integer and $H$ is a subgroup of $\mathbb Z_n$. Prove that either every member of $H$ is even or exactly half of members of $H$ are even. Same question have been asked ...
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35 views

Show that in a group order of element is less than or equal to order of group [closed]

Show that in a group order of element is less than or equal to order of group. This is a Question from Gallian. Please provide hints on how to start this? THanks
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2answers
57 views

Prove that if $G$ is cyclic and infinite then $G$ is isomorphic to $\mathbb{Z}$

Assume $G$ is generated by $a$, so $G = \langle a\rangle$. Since $G$ is infinite for all $m \in \mathbb{Z}$, $a^m \neq e$. Suppose $a^h = a^k$ then $a^h\cdot a^{-k} = a^{h-k} = e$, but this is a ...
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43 views

The software for finite field arithmetic

Is there any software, library,or toolkit that support arithmetic with normal basis on $GF(2n)$ field? What is the best one? Especially, which software can implement the conversion between normal ...
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27 views

Prove the Radical of an Ideal is an Ideal

I am given that $R$ is a commutative ring, $A$ is an ideal of $R$, and $N(A)=\{x\in R\,|\,x^n\in A$ for some $n\}$. I am studying with a group for our comprehensive exam and this problem has us stuck ...
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43 views

Units of $\mathbb{Z}[\sqrt{7}]$

Let $\mathbb{Z}[\sqrt{7}]=\{a+b\sqrt{7}\mid a,b\in\mathbb{Z}\}$. Let $\mathbb{Z}[\sqrt{7}]$ have the usual addition and multiplication, namely $$(a+b\sqrt{7})+(c+d\sqrt{7})=(a+c)+(b+d)\sqrt{7}$$ and ...
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37 views

What is the stalk of the structure sheaf of the plane?

Let $\mathcal{O}$ be the structure sheaf of $\mathbb{A}^2_\mathbb{C}$. How do I compute the local ring corresponding to the stalk of $\mathcal{O}$ of the point $(0,0)$? I tried computing the ...
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2answers
54 views

Show that open interval $(-1,1)$ is isomorphic to $(\mathbb{R},+)$

Define group structure on $G=(-1,1)$ by $$a*b=\frac{a+b}{1+ab}$$ for any $a,b\in G$. Show that $G$ is isomorphic to $\mathbb{R}$ under addition. I've tried the obvious maps $f:G\rightarrow ...
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Show that $\left\| \exp(A)-\mathbf{1} \right\| \leq e^{\left\|A\right\|}-1$

Have been attempting this question, just wondering if my answer looks alright. Question: Given $A \in \Bbb{K}^{n\times n}$ show that $\left\| \exp(A)-\mathbf{1} \right\| \leq e^{\left\|A\right\|}-1$ ...
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subgroup of a semidirect product

I'm really lost with this problem and I really need your help: Let $G=\mathbb{Z}^2\rtimes_A\mathbb{Z}$, and let $H\leq G$ with finite index in G. I have to prove that there is a subgroup $U$ of ...
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40 views

Let $F\subset E$. If $f(x)$ is solvable over $F$, prove that $f(x)$ is solvable over $E$. [closed]

I'm trying to show that if $f(x)$ is solvable over $F$, then $f(x)$ is also solvable over $E$. It's intuitively true, but I don't know how to show it rigorously.
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Proving that the index of a subgroup of $S_n$ that keeps a specific set invariant has a certain order

Let $n \in \mathbb{N}, n ≥ 2$, and $k \in \{1, 2, ..., n-1\}$, and let $A \subseteq \{1, 2, ..., n\}$ with $|A| = k$. Furthermore, let $G$ be a subgroup of $S_n$ that fixes $A$, i.e. for all $π \in ...
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1answer
56 views

Automatic additivity of multiplicative maps

There are results which guarantee that a multiplicative bijection between commutative rings is actually additive (that is, it is a ring isomorphism). For example this result of Martindale initiated ...
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1answer
38 views

If a finite group $G$ has a normal subgroup $N$ of order $p$ where $p$ is the smallest prime divisor of the group order, then $Z(G)$ is nontrivial

Let $G$ be a finite group and $p$ the smallest prime number with $p \mid |G|$. I want to show: if $G$ has a normal subgroup $N \trianglelefteq G$ of order $p$, then $G$ has a nontrivial center. My ...
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1answer
48 views

Fundamental theorem of Galois Theory problem

Let $E/F$ be a Galois extension of degree $p^k$. Prove that there exists an intermediate field $K$ with $[E:K] = p$ and $K/F$ Galois of degree $p ^{k−1}$. I think I know how to prove the former but I ...
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40 views

Finding the Left and Right Cosets in $A_4$

I'm really struggling with a Group theory class and would love some help. HW Question is as follows. Consider the subgroups $H = \left<(123)\right>$ and $K=\left<(12),(34)\right>$ of ...
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1answer
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Calculating the Matrix of a Transformation Using Bases of Field Extensions

I'm trying to understand this topic in my Abstract Algebra class: Suppose that we have a finite field extension $L/F$ and let us choose $a \in L$. We'll define the transformation $T_{a} : L \to L$ ...
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56 views

Unramified algebraic extensions of local fields

This is a basic question from Neukirch's Algebraic Number Theory, Prop. 7.2: Fix a non-Archimedean local field $K$. Let $L/K$ and $K'/K$ be two extensions inside an algebraic closure $\bar{K}/K$ and ...
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1answer
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If a is a group element and a has infinite order, prove that $a^m\neq a^n$ when $m \neq n$

If $a$ is a group element and $a$ has infinite order, prove that $a^m\neq a^n$ when $m \neq n$. (Gallian, Contemporary Abstract Algebra, Exercise 19, Chapter 3.) To prove that $m \neq n$ implies ...
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In a group we have $a^6=e$. What are possibilities for order of a? [duplicate]

In a group we have $a^6=e$. What are possibilities for order of a? I think order of a can be 2 or 3. Because if order of a is 2. Then $a^2=e$. Multiplying by $a^4$ we get $a^6=e$ which is true. ...
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Extension $L/K$ and the field $K[a]$

I'm not sure I fully understand what is an extension $L/K$. Is it correct to say it is a field $L$ that contains a subfield isomorphic to $K$? Keeping this in mind, is it correct to say that ...
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Why do we introduce groups using division?

I am only starting to really study algebra so I apologize if this is an ill-formed question. When learning about groups, why is division used so heavily in the beginning? Would it not be simpler to ...
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1answer
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Cyclic algebras of degree $4$ and period $2$

Recall that if a field $k$ has a primitive $n$-root of unity $\omega$, then the cyclic $k$-alegbras of degree $n$ (ie of dimension $n^2$) have the following familiar presentation : they are generated ...
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1answer
37 views

Find a primitive element in the splitting field of $x^4-8x^2+15$.

I'm trying to find a primitive element in the splitting field of $x^4-8x^2+15$. I don't know in general what should I do with this kind of questions. Should I solve the roots and get the Galois group? ...
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1answer
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Is every such family induced by an associative operation?

Suppose we're given an associative operation $\star : X \times X \rightarrow X$. Then for each $n \in \mathbb{N}_{>0}$, there's a function $f_n : X^n \rightarrow X$ given as follows: ...
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Find the Galois group of $x^4-x^2-6$.

I'm trying to find the Galois group of $x^4-x^2-6$. I think there are 4 roots, thus I guess the Galois group is $A_4$? But I don't know in general how to solve this.
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Self conjugation on a group with $2p$ elements

Let $G$ be a group with $2p$ elements where $p$ is an odd prime. Also let $Z(G)=e$, where $e$ is the identity element. Prove that there is a conjugationclass with $p$ elements. My attempt: Because ...
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1answer
34 views

Image of ring homomorphism is a subring

Let $\phi : R\to S $ be a ring homomorphism. Definition: $Im\left ( \phi \right )=\left ( R \right )\phi=\left \{ \left ( a \right )\phi \mid a \in R \right \}$ Prove that $Im\left ( ...
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If polynomials are almost surjective over a field, is the field algebraically closed?

Let $K$ be a field. Say that polynomials are almost surjective over $K$ if for any nonconstant polynomial $f(x)\in K[x]$, the image of the map $f:K\to K$ contains all but finitely many points of $K$. ...
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Ring homomorphism of polynomial ring

Let $R\left [ x \right ]$ be a Polynomial ring. Let R be a ring If $R\left [ x \right ]\rightarrow R$ $f\left [ x \right ] \mapsto f\left ( 0 \right )$ is a ring homomorphism ...
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1answer
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Conjugacy classes in topological groups are closed?

EDIT Just realized that this question Conjugacy classes of a compact matrix group is related, but I think that the answer use specific properties of matrix groups, so it doesn't apply. QUESTION ...
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Prove that the system $(P, S, 0)$ satisfy Peano Axioms.

Peano Axioms. Let $\mathbb N \neq \emptyset$ be a set and $S:\mathbb N \to \mathbb N$ a function. The elements of $\mathbb N$ are the natural numbers. If $n \in \mathbb N$ then $S(n)$ is the ...
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1answer
34 views

Proving that polynomial is irreducible over F and classifying its roots in a splitting field

Let $F =\mathbb F_{2}(u)$ be the field of rational functions over the prime field $\mathbb F_{2}$. Prove that $x^2-u$ is irreducible over $F$ and that it has a double root in a splitting field. ...
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Is this an equivalent statement to the Fundamental Theorem of Algebra?

Is the following equivalent to the usual statement of the fundamental theorem of algebra: Let $$f(z)=c_nz^n+\cdots+c_1z+c_0$$ be a polynomial with complex coefficients. For all but finitely many ...
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30 views

Name of a family of Coxeter groups

From the following image I know that the first of group is the symmetric group of rank $n$ and the second is known as the Hyperoctahedral group. I want to know if someone knows the name of the ...
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3answers
52 views

Do you write plus/minus if a variable squares equals the square root of a number?

For example, if I have $x^2 = \sqrt{49}$. I know that $7$ is the number, but as my final answer, do I write that $x = +\sqrt{7}$ and $-\sqrt{7}$ or just $x = \sqrt{7}$?
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Hall $\pi$-subgroups of a normalizer

Let $G$ be a group with $A\unlhd G$ and $H\in$ Hall$_\pi$(G) Consider $N_G(H \cap A)$. Then $H$ is a Hall $\pi$-subgroup of $N_G(H \cap A)$ Let $K$ be another Hall $\pi$-subgroup of $N_G(H \cap A)$ ...
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f is null homotopic if and only if (-s,f):cone(C)->D

Actually this question is from Weibel, exercise 1.5.2. Let $f:C\to D$ be a map of complexes. Show that $f$ is null homotopic if and only if $f$ extends to a map $(-s,f):$cone($C$)$\to D$. ...
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1answer
33 views

Degree of a finite field extension

Let $i,\sqrt{3}\in\mathbb{C}$. I know that both are algebraic over $\mathbb{Q}$. Hence $[\mathbb{Q}(i\sqrt{3}):\mathbb{Q}]=\deg(i\sqrt{3},\mathbb{Q})$. This is equal to 2 since ...