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 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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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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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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Every group element is a product of elements in certain subsets

Let $G$ be a group. For $\theta \in$ Aut$(G)$ of order $2$, define $$ K:=\{ g\in G \mid \theta(g)=g \},\quad S:=\{ \theta(g)^{-1}g \mid g\in G \}.$$ My first question is: Assume there is a ...
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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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Field Trace/Norm and Matrix Trace/Norm (Dummit and Foote 14.2.31(c)).

I can't quite figure out this final part to 14.2.31 in Dummit and Foote, 3rd edition. I'm given $K/F$ is a finite field extension of degree $n$, and $\alpha\in K$. I've shown that the map ...
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33 views

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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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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49 views

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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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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68 views

Show that the permutation $(1 \space 2 \space 3)$ can not be a cube of any element of $S_n.$

Here is my try: If there exists $a \in S_n$ such that $a^3=(1 \space 2 \space 3)$, then $a^9=e$ where $e$ is identity in $S_n$. Then $o(a)=9$. I don't know how to proceed further. Can anyone ...
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1answer
32 views

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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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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Question about the rational normal curve and different representations of it.

I know the rational normal curve as the image of a polynomial map \begin{gather} \phi:K\rightarrow K^n\\ \phi(t)=(t,t^2,\dots,t^n) \end{gather} My question is proving the variety defined by the set ...
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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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29 views

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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23 views

Number Theory & Abstract Linear Algebra [closed]

How to prove that if a, b are co-prime to a positive integer n, then (ab mod n) is also co-prime to n.
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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
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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abelian tower being solvable

While I am reading serge lang's algebra, I am confused by the definition of solvable group. In the book, $G$ is solvabe if $G$ has abelian tower with last element being trivial subgroup. Is he ...
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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 ...
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Hopfian and Co-Hopfian Modules.

Let $M$ be a $R-$module. We say that: $M$ is a Hopfian module, if every epimorphism of $M$ is a monomorphism. $M$ is a Co-Hopfian module, if every monomorphism of $M$ is an epimorphism. Why do we ...
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33 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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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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Prove that a free group of rank $\ge2$ is centerless and torsion-free

This exercise is from Rotman's Introduction to the Theory of Groups. It's just as the title states: prove a free group of rank $\ge2$ is centerless torsion-free. Here, the definition of a free group ...
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Equation over free group

Let us consider free group $F(a,b)$ of rank two. I need to find a solution (or to prove that there is no one) over this group of the following equation: $$x^2[x^{2k},y]=a^2b^2,$$ where ...
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Great Circles in $SU_{2}$

So I am working on the proof that all great circles in $SU_{2}$ (circles of radius 1) are a coset of a longitude, and I am unsure what a great circle looks like in matrix form. Clearly any point on ...
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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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Counting homomorphisms of groups

How many homomorphisms are there from $A_{5} × S_{3} × A_{4}$ to $\mathbb Z/2\mathbb Z$? I know that $Z/2Z$ is generated by ome element and of order two. I know the respective order of the rest ...
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1answer
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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 ...
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Determining the size of a subgroup of H

Given that $|GL_{2}(Z_{5})|=480$ find the index of $H$ in $GL_2(\mathbb{Z}_5)$. $$ H=\begin{pmatrix} a & b \\ 0 & c \\ \end{pmatrix} \\a,c \neq 0 , \\a,b,c\in \mathbb{Z}_5 $$ I proved in ...
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If $v$ is algebraic over $K(u)$, for some $u\in F$, and $v $ is transcendental over $K$, then $u$ is algebraic over $K(v)$

If $v$ is algebraic over $K(u)$ for some $u\in F$, $F$ is an extension over $K$, and $v $ is transcendental over $K$, then $u$ is algebraic over $K(v)$. I came across this problem in the book Algebra ...
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Killing form is negative definite

Let $G$ be a compact connected semisimple Lie group and $\frak g$ its Lie algebra. It is known that the Killing form of $\frak g$ is negative definite. What about the Killing form $B$ of the complex ...
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Faithful representation of a $p$-group

Suppose $G$ is a nontrivial $p-group$. Let $H$ be the intersection of the center of $G$ and the set of elements in $G$ of exponent $p$. Let $\rho: G\rightarrow GL(V)$ be a representation. Show that if ...
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Showing a subset $K$ is a subfield of a field

Let $F$ be a field and let $K$ be a subset of $F$ with at least two elements. Prove that $K$ is a subfield of $F$ IF, for any $a,b$ ($b\neq 0$) in $K$, $a-b$ and $a\cdot b^{-1}$ belongs to $K$. ...
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Finding Galois Group

Let $\mathbb Q\subset\mathbb Q(\sqrt{3}+\sqrt[3]{5})$. I want to find the Galois group of the given field extension. It would be easy for me if I could find a basis of the given field extension ...
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Let $\mathbb Z[\omega] = \{x + \omega y | x, y \in \mathbb Z\}$ where $\omega^2 + \omega + 1 = 0$ Prove that $N(z) = z\bar z = x^2 - xy + y^2$

Let $\mathbb Z[\omega] = \{x + \omega y | x, y \in \mathbb Z\}$ where $\omega^2 + \omega + 1 = 0$ Let $x+ \omega y \in \mathbb Z[\omega]$ and let $\bar z$ denote the complex conjugate of $z$. Prove ...
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Let $A$ be a ring, $M''$ and $P$ be $A$-modules and $f:P\rightarrow M''$ a homomorphism…

I need some help with this problem, it seems easy, but I don't get it. Let $A$ be a ring, $M''$ and $P$ be $A$-modules and $f:P\rightarrow M''$ an homomorphism. Suppose that for each $A$-modules $M$ ...
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$\gcd(a, 63) = 1$ implies $a^7 \equiv a \mod 63$?

Let $a$ be an integer. Suppose that $\gcd(a,63) = 1$. Prove then that $$a^7 \equiv a \mod 63. $$ Attempt: Since $gcd(a,63) = 1$, by Fermat little theorem we have that $$a^{\phi(63)} \equiv 1 \mod ...
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Is $\mathbb{Z}$ contained in $\mathbb{Z}_p$? [closed]

I was wondering if $\mathbb{Z}$ is contained in $\mathbb{Z}_p$, the group of integers modulo $p$? As I can take every integer and send it to it equivalence class I believe that I could be possible?
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*-algebra Morphisms

Let $A$ and $B$ be two $\ast$-algebras and let $\varphi:A\to B$ be a $\ast$-algebra morphism. I am interested in all the ways that $\varphi$ could fail to be unital (that is, if $A$ has a unit $1_A$, ...
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The rigid additive tensor category freely generated by an object

I've found myself to be absolutely mystified by something in Deligne and Milne's notes on Tannakian categories. Namely, on p. 16 they are showing that there is a rigid additive tensor category ...
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How many points at infinity in Artin-Schreier type curve

Let $Y$ be an affine curve over a perfect (yet not necessarily algebraically closed) field $k$ given by $$y^p+a(x)y=b(x)$$ (abs. irreducible) with $p$ a prime number. Now one can normalize $k[1/x]$ in ...
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1answer
34 views

What is a basis of the polynomials in two variables $X_1, X_2$ over K as K-algebra?

I am trying to undertsand the notion of a basis when working with K-algebras. Am I right that a K-algebra would be generated by $1,X_1$ and $X_2$? I read somewhere that it is generated by $X_1$ and ...
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Proving equivalent formulations of Jacobson radical

I should start by saying I found this post Equivalent definitions of the Jacobson Radical which is about the same two formulations of the Jacobson radical but it didn't really to answer my question. ...
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Localization and completion under a strong hypothesis

This question is closely related to this one, but in my case I think the hypotheses are different. Let $(A,\mathfrak m)$ be a regular, local noetherian domain (the local ring at a smooth point of ...