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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A question on part of the proof of the theorem that if $R$ is a UFD then $R[x]$ is a UFD as well.

I have a question regarding a proof in Peter Falb's Methods of Algebraic Geometry in Control Theory, volume I, for the claim in the title. On pages 16-17 he proves the property (ii) of UFDs that the ...
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Problem sets on Abstract Algebra

Many times we ask about what books should we read to learn or know more about a math topic (Abstract Algebra, in this case). However, I would like to get a list of the exercises what should we solve ...
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Name for submodule killed by a right ideal

Let $\mathfrak a$ be a right ideal in a ring $R$. The set $N=\{m\in M: \mathfrak am=\mathfrak 0\}$ is a submodule of the left $R-$module $M$: If $m,n\in N$, $a\in \mathfrak a$, then $a(m-n) = am-an =...
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Can the dihedral groups be seen as subgroups of each other?

I am solving one difficult problem and now I need information on this: If $m\mid n$, is then possible to "embed" $D_m$ to $D_n$, or otherwise said, does $D_n$ have a subgroup isomorphic to $D_m$? ...
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261 views

Algebraic structure on any infinite set

Given any algebraic object $X$, say group, ring, integral domain, etc., and a special subset $I$ of $X$ namely normal subgroup, ideal etc., it is always possible to put a structure on $X/I$ induced ...
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Quotient ring with reducible polynomial

Let $S = \mathbb{R}[x]/(x^2+1)^2).$ The first goal is to show that there exist exactly two homomorphisms $ \pi\colon S\to \mathbb{C} $ such that $\pi|_{\mathbb{R}} = \text{id}_{\mathbb{R}}.$ I know ...
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Factoring $p(x) = x^n -1$ for any natural number $n$

Can I say that from inspection, $(x-1)$ is a factor which implies that $p(1) = 0$. I then used long division which then gave this: $p(x) = (x-1)(\sum \limits _{i=1} ^{n} x^{n-i})$ How would you have ...
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I am confused about this notation for matrix representation.

I am confused about this notation from the image below. How can you represent something like (1,2),1 as a 2D matrix? For S1 = (0,1,2) and S2 = (0,1) I would expect two matrices like: [(1,0),(1,0),...
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$17\mathbb{Z}[\sqrt {10}]$ is prime ideal in $\mathbb{Z}[\sqrt {10}]$

This seems tedious since I would have to show $(a+b\sqrt {10})(c+d\sqrt {10})=17k+17j\sqrt {10}$ implies that one of the factors belongs to $17\mathbb{Z}[\sqrt {10}]$. It's easier to show something ...
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69 views

Show that $V=\mathbb KG\oplus\cdots\oplus \mathbb KG$.

Let $V$ a $\mathbb KG$-module such that the character of $V$ is such that $\chi_V(g)=0$ for all $g\in G\setminus \{1\}$. Show that there is an $m$ s.t. $$V=\underbrace{\mathbb KG\oplus\cdots\oplus \...
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Sylow subgroup of a product of subgroups

Let $UV$ be a subgroup of $G$ and $P$ is a Sylow $p$-subgroup of $G$. Suppose that $P \cap U \in \operatorname{Syl}_p(U)$ and $P \cap V \in \operatorname{Syl}_p(V)$. I need to show that $P\cap UV \in \...
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Automorphism of $\mathbb C(x,y)$ and its order in $\mathrm{Aut}(\mathbb C(x,y))$

In the following problem Let $M=\bigg (\begin{matrix} a& b\\ c& d\end{matrix}\bigg )$ be a nonsingular matrix with integer coefficients and $L=\mathbb C(x,y)$. (i) Show that $\phi(x)=...
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Properties of a subgroup of a group $\mathbb Z_p \times \mathbb Z_p$

Let $p \geq 5$ be a prime. Thhen which one of the followings are true. 1) $\mathbb Z_p \times \mathbb Z_p$ has atleast five subgroup of order p. 2) Every subgroup of $\mathbb Z_p \times \mathbb Z_p$ ...
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36 views

Character group and lattices

Let $\Lambda$ be the complex n-th dimensional lattice over Eisenstein integers ($\mathbb{Z}[\omega]$)). The map $R: \mathbb{C} \mapsto \mathbb{R}$ is defined as following: $R(z)=R(z_{a}+\omega z_b)=...
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28 views

Finding an essential submodule

Let $R$ be a commutative ring with unity, and let $M$ be a unitary faithful $R$-module. Assume the annihilator $A$ of $r\in R$ is essential in $R$ (as an $R$-module). I search for an essential ...
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42 views

A sum of powers of primitive roots of unity

For the primitive roots of unity $\omega_n = e^{i2\pi/n}$ I want to prove that $$\sum_{k=0}^{n-1} \omega_n^{lk} = 0$$ if $n$ doesn't divide $l$. I have already proven the well-known result $$\sum_{k=...
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55 views

Examples of irreducible polynomials over a finite field with prescribed coefficients [on hold]

I came to know that it is an open problem, but I am not able to find any simple example to explain it properly. Can some one help me with some simple explanation regarding what this problem is about?...
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113 views

Specific basis of A-algebra B that is also a free A-module of finite rank.

I have a problem that seems (at least to me) harder then I initially thought. Let $B$ be an $A$-algebra that is also a free $A$-module of finite rank (if necessary we can assume that $B$ is ...
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$\mathbb{Z}[\sqrt{-5}]$ satisfies the descending chain condition of divisors

I want to show that $\mathbb{Z}[\sqrt{-5}]$ satisfies the descending chain condition of divisors: given a chain $a_1,a_2,\dots,a_n,\dots$ and $a_{n+1}\mid a_n$ for any $n\in \mathbb{N}$, then there is ...
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$I$ is the maximal left ideal

Let $R$ be a ring and $I\subseteq R$ the unique maximal right ideal of $R$. I have shown that $I$ is an ideal and that each element $a\in R-I$ is invertible. I want to show that $I$ is the unique ...
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Let $M$ be a finitely generated module over a P.I.D.. If $M=A\oplus B$, $A$ is a torsion submodule and $B$ is free, then $A=M_{\text{tor}}$.

I am reading the Goodman's Algebra. There is a lemma. Let $M$ be a finitely generated module over a principal ideal domain $R$. (a) If $M=A\oplus B$, $A$ is a torsion submodule, and $B$ is free, ...
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Is $a$ invertible?

We have that $R$ is a ring. Suppose that $Ra=R$ and $bR=R$, for $a,b\in R$. Then we have that there is $x\in R$ such that $ab=1$ and $bx=1$. Does it follow from that that $a$ is invertible?
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Cardinallity increasing constructions

If we start with $\mathbb{Z}$ we can through localization get $\mathbb{Q}$, but that has the same cardinallity as $\mathbb{Z}$, so it doesn't increase cardinality for infinite sets, which is what I am ...
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Is an algebra homomorphism between two finitely generated algebras over a field automatically an integral morphism?

I'm having a bit of trouble with the idea of an integral morphisms, and algebra homomorphisms for that matter. I'm wondering if the above is just "automatically" true. Does an algebra over a field ...
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40 views

What are the invertible elements of the $K[X]$ ring where $K$ is a field? [duplicate]

What are the invertible elements of the $K[X]$ ring where $K$ is a field? I know that in a field all the elements are invertible but I'm having trouble linking $K$ and $K[X]$
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65 views

Image drawing complex analysis [on hold]

$w=u+iv,z=x+iy$ are complex numbers and we have $w=z^2-2z$. Determine the image in the $w$-plane of the unit circle $x^2+y^2=1$. I have tried to answer this here Question and Answer. I have problems ...
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Find a splitting field of $x^2 + 1$ over $\mathbb{Z}_3$

We know that $x^2 -3$ is irreducible in $\mathbb{Q}[x]$. We also know that $\sqrt{3}$ solves $x^2 - 3$. As a result, $x^2 - 3 = (x-\sqrt{3})(x+\sqrt{3})$. This implies that the splitting field of $x^2 ...
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If $R\otimes_\mathbb R\mathbb C$ is finitely generated $\mathbb C$ - algebra then $R$ is a finitely generated $\mathbb R$ - algebra?

Let $R$ be an $\mathbb R$ - algebra. Suppose $A=R\otimes_\mathbb R\mathbb C$ is a finitely generated $\mathbb C$ - algebra then is $R$ a finitely generated $\mathbb R$ - algebra? I thought along the ...
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Simple examples of rings from topology

The ring $C([0,1],\mathbb{R})$ of continuous functions from $[0,1]$ to $\mathbb{R}$ is an interesting example of ring due to its some interesting property (namely, structure of maximal ideals). The ...
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Field of fractions of ring F[x] [on hold]

Let $F$ be a commutative ring without zero divisors and $Q$ its field of fractions. Prove that field $Q(x)$ is a field of fractions of ring $F[x]$ Thanks for any help.
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Legendre symbol $(-21/p)$

I am a bit confused with the question: For what prime $p$, $\left(\frac{-21}{p}\right) = 1$? I did something like that: $$\left(\frac{-21}{p}\right) = \left(\frac{-1}{p}\right)\left(\frac{3}{p}...
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Which functions $\mathbb{Z} \rightarrow \mathbb{Z}$ are 'totally compatible'?

Definition 0. For each integer $k$ and each function $f : \mathbb{Z} \rightarrow \mathbb{Z}$, lets define that $f$ is $k$-compatible iff there exists a function $g : \mathbb{Z}/k\mathbb{Z} \rightarrow ...
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In transitive (non-trivial) group action, there must be at least one group element without fixed point

Let a finite group $G$ act transitively on a finite set $S$ with $|S| \geq 2$. The problem is to show that not every $g \in G$ can have a fixed point in this action. I proved this on my own, but I'm ...
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prove Euclidean $ℚ^p$ $ℚ_p$ [on hold]

prove Euclidean 1) $ℚ^p$ with norm $n(kp^m) = |k|$, $k,m ∈ ℤ$, $(k, p) = 1$; 2) $ℚ_p$ with norm $n(\frac{a}{b}p^m) = p^m$, $a,b,m ∈ ℤ,$ $m≥0$ $(ab, p) = 1$;
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1answer
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Find the minimum, maximum, minimals and maximals of this relation

Tell if the following order relation is total and find the minimum, maximum, minimals and maximals: $$\forall a,b \in\mathbb Z,\ \ a\ \rho\ b \iff a \leq b\ \text{ and }\ \pi(a) \subseteq \pi(b)$$ ...
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How to show $\sqrt{3} + \sqrt{2}$ is algebraic? [duplicate]

How to show $\sqrt{3} + \sqrt{2}$ is algebraic? Is there a way I can do this without trial and error? Thanks.
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1answer
35 views

Algebra problem about Ker and Im

I have a problem with this linear algebra exercise. A) Find an orthonormal basis with respect upon the Euclidean product for a vector space in $\mathbb{R}^3$, generated by those vectors: $(1, 2, -1)$...
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Interpretation of the join of two stabilizer subgroups

Let $G$ be the group acting on two sets $X, Y$. Let $G_x$ and $G_y$ be stabilizer subgroups of some elements $x \in X, y \in Y$. It is easy to see that $G_x \cap G_y = G_{(x,y)}$, when we combine two ...
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Which one of these two is an equivalence relation

I'm having an issue with the following exercise: Given $\alpha$ and $\beta$ two binary relationships defined in $Z$ such that: $$\forall m,n \in Z, n\ \alpha\ m \iff n = m\ \ \vee\ \ rest(n,7)\ +...
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A binary operation, closed over the reals, that is associative, but not commutative

I am aware that matrix multiplication as well as function composition is associative, but not commutative, but are there any other binary operations, specifically that are closed over the reals, that ...
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If $a$ has order $p$ and $aPa^{-1}=P$ then $a\in P$

If $G$ is a finite group, $P$ a $p$-Sylow subgroup of $G$, $a\in G$ has order $p$ and $aPa^{-1}=P$ then $a\in P$. This is proved in Rotman: Proof: We have $a\in N_G(P)$. If $a\notin P$ then $aP\in ...
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Number of subrings of a Field

let $p$ be a prime number; then how many distict subrings (with unity) of cardinality $p$ does the field $F_{p^2}$ have? 1.$0$ 2.$1$ 3.$p$ 4.$p^2$ I think it will have only one subring of ...
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Different right / left identity and two sided identity element

Maybe this will be a trivial question but I can't find a solution for this. I checked already on a lot of questions here and on google but can't find a solution for my specific problem. Given an ...
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Question about multiplicative arithmetic functions

Let $f,g:\Bbb N\to \Bbb C$ be multiplicative arithmetic functions, i.e. $$\gcd(m,n)=1\implies f(mn)=f(m)f(n)$$ and same for $g$. We can also assume $f(1)\neq 0$ and $g(1)\neq 0$ if necessary. How can ...
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Question about center of a group/subgroup.

Let $G$ a non abelian group of order $p^aq^b$ where $p,q$ are prime and $p^aq^b$ is not prime. I want to show that $G$ is not simple. In a proof of my course, they do as following for the case where ...
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38 views

Showing an isomorphism of rings

Consider the ideal $I=(1+2x)\cdot \Bbb Z[x]$ in the polynomial ring $\Bbb Z[x]$. I am trying to show that $\Bbb Z[x]/I$ is isomorphic to $R=\{\frac{a}{2^r}:a\in \Bbb Z, r\in \Bbb N_0\}$. My approach:...
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Non-algebraic structures?

We call group, ring, field,... "algebraic structures". Do we have similar analogue for transcendental numbers? If not, then how do we study interactions between various transcendental numbers? Also, ...
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Factor ring and prime elements

My task is to find prime elements of the ring $\mathbb Z[\sqrt{-21}]$ and describe the factor ring $\mathbb Z[\sqrt{-21}]/(2+\sqrt{-21})$. I think that to describe factor-ring i need to find the ...
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24 views

Finitely generated Ideals of finite algebras

i would really appreciate any help with this question. So, the question is: How to prove that finitely generated ideals of finite algebras over the ring F are finite over F? Thanks.
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What is the minimum and maximum of a set with only one element?

This is surely a trivial question but I want to be sure I understand correctly what happens. Given a set $A = \{1\}$, what is $\min A$ and $\max A$? Is it $\min A = 1$ and $\max A = 1$?