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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$\mathbb{Z}_p$ is an Integral Domain

Assume that we define the ring of the $p$-adic integers as the projective limit $$\mathbb{Z}_p =\varprojlim \frac{\mathbb{Z}}{p^n\mathbb{Z}}$$ Then $\mathbb{Q}_p$, the field of the $p$-adic numbers ...
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Given a homomorphism defined on a generating set of a group how to define it for a general element?

Let $\mathbb Z^n$ and $\mathbb Z^d$ be free $\mathbb Z$-modules with $d>n$. Suppose $v_1,\dots,v_d$ are primitive vectors in $\mathbb Z^n$. Let $e_1^*,\dots, e^*_n$ be the dual basis (that is, a ...
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Number of subgroups equal to order of group

Here is a fun question. Consider the dihedral group $\mathcal{D}_4=\left\langle a,b\mid a^4=b^2=1, bab=a^{-1}\right\rangle$ of order $8$. This group has exactly $8$ genuine subgroups (but not all ...
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Are there groups $G$ isomorphic to $\mathrm{Aut}\left( \mathrm{Aut} (G)\right)$, such that $G\not\cong \mathrm{Aut}(G)$?

Trivial examples for the first condition are easy to find: $G={1},C_2$. Are there any (finite\non-finite) groups that satisfy the conditions in the title?
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Finding square root of polynomial in extension field (Number field sieve)

I was reading this paper, on page number 29, 2nd paragraph it is written that "take the coefficients of $ \gamma $ modulo q and applying an algorithm for taking square roots in the finite field Z[ $ \...
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71 views

Homology Groups of Tangent 2-Spheres

I have been trying to compute the Homology Groups $ H_n $ of two tangent 2-Spheres (we will call this space, X). By previous results, I am able to easily determine that $ H_0(X) $ is isomorphic to the ...
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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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21 views

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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41 views

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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49 views

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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38 views

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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2answers
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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32 views

$\mathbb{Z}[\sqrt{-5}]$ 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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47 views

$I$ 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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1answer
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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1answer
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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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61 views

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] [closed]

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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63 views

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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35 views

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 $ℚ^p$ $ℚ_p$ [closed]

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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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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33 views

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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35 views

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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1answer
88 views

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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2answers
33 views

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 ...