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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Matrix $B \in M_n(S)$, for $S$ an $R$ algebra, with $R$ independent entries, $A \in GL_n(R)$. Are the entries of $AB$ $R$-independent?

Let $R$ be a field (or a domain, or a commutative ring), and $S$ an $R$-algebra. Let $B \in M_n(S)$ have $R$ independent entries. Let $A \in GL_n(R)$. Are the entries of $AB$ $R$-independent? I am ...
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Polynomial-closed properties of rings

If $R$ is a ring with certain property, sometimes when we pass to the polynomial ring in one variable, the ring $R[x]$ still has the same property. For instance, it's a theorem that if $R$ is a UFD ...
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Some properties of finite subgroup G of multiplicative group $F^*$

F is a field. $\psi_G(d)$ is number of elements with order d. N(F)- set of all zeros of polynomial $X^d-1 \in F$ and $|G|=m$. For $d\in \mathbb{N}$ and $\psi_G(d)\neq 0$ Show that: $d|m$ and ...
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1answer
14 views

Verify that $A \oplus B$, where $A$ and $B$ are cyclic groups of orders 2 and 3, is the cyclic group of order 6

Let's define $A$ and $B$ as follows: $A$ = {e,a} $B$ = {e,b,2b} Then $A\oplus B= \{\{e+e\},\{e+b\},\{e+2b\},\{a+e\},\{a+b\},\{a+2b\}\}$ which is equal to ...
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1answer
49 views

Aut$(G)\cong \Bbb{Z}_8$

I am looking for a group such that Aut$(G)\cong \Bbb{Z}_8$. Obviously Aut$(\Bbb{Z}_n)\ncong \Bbb{Z}_8$ for any $n$. Also Aut$(D_4)\cong D_4$, neither symmetric/alternating groups are of any help ...
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25 views

Set of functions from a finite field to the integers

Has the set of functions $\mathbb{F}_q \to \mathbb{Z}$ endowed with pointwise addition and additive/multiplicative convolution been studied? Does anyone know a reference or keywords to search for? ...
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19 views

*$G$-invariant* symmetric bilinear form & $G'=\Bbb Z_2\times\Bbb Z_2$.

I got a problem with the last point I solved all the points, from (a) to (h), but I have no idea how to solve (i): how can I associate a bilinear form to a represtation? What is a $G$-invariant ...
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2answers
20 views

Basis of a field extension

Let $K$ be a field, and let $A$ be a $K$-algebra such that $\alpha \in A$. Then the natural homomorphism $$ \phi: K[x] \to K[\alpha], \hspace{3mm} (x \mapsto \alpha )$$ has a kernel which is a ...
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42 views

General notions of basis

Free groups, free abelian groups, and vector spaces all have a notion of 'basis': a subset $B$ of the structure such that everything in the structure can be written uniquely as a finite combination of ...
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1answer
59 views

Under what conditions does $M \oplus A \cong M \oplus B$ imply $A \cong B$?

This question is fairly general (I'm actually interested in a more specific setting, which I'll mention later), and I've found similar questions/answers on here but they don't seem to answer the ...
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$\operatorname{Hom}_R(\mathfrak{a},M)$ is isomorphic to $\mathfrak{a}^{-1}M$ if $R$ is a Dedekind domain

I want to prove Lemma 2.5.1 of Silverman's "Advanced Topics in The Arithmetic of Elliptic Curves" (whose proof is left to the reader): "Let $R$ be a Dedekind domain, let $\mathfrak{a}$ be a ...
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1answer
25 views

Power series ring $k[[x]]$ contains elements transcendental over $k(x)$

If $k$ is countable, then $k[x]$ is countable and it seems easy to figure out $k[[x]]$ has elements transcendental over $k(x)$ because $k[[x]]$ is uncountable by using the fact that algebraic closure ...
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49 views

Example of $A$-module but not $A$-algebra. [duplicate]

If $A$, $B$ are commutative rings, and if $B$ is an $A$-algebra then it is also an $A$-module. I am looking for an example that shows that the converse is not true. That is, I am looking for ...
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Homomorphisms from $\mathbb{C}$ to $M_2(\mathbb{R})$

Let $\phi_1$ and $\phi_2$ be two injective ring homomorphisms from $\mathbb{C}$ to $M_2(\mathbb{R})$. Show that there exists a $g\in GL_2(\mathbb{R})$ such that $\phi_2(x) = g\phi_1(x)g^{-1}$ for all ...
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1answer
43 views

What is the difference between submodules of $A/\mathfrak a$ as an $A$-module or as an $A/\mathfrak a$-module?

If $\mathfrak a$ is an ideal of unital commutative ring $A$, Then we can see to $A/\mathfrak a$ as an $A$ module or as an $A/\mathfrak a$ module. If $A=\mathbb Z$ there is no structure difference ...
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44 views

Show that every short exact sequence $0\rightarrow M' \rightarrow M \rightarrow M'' \rightarrow 0$ (with $M''$ free) is split exact

Note: $M,M',M''$ are modules. In order to show that it's split exact, I understand that I need to show that $\beta:M\rightarrow M''$ has a left inverse, and similarly that $\alpha:M'\rightarrow M$ ...
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recover (pontrjagin) ring structure from the localization (w.r.t. $\pi_0$)

Let $R$ be a ring and $S$ a given multiplicative subset of $R$. Suppose we know the multiplication structure of $S$. If we know the ring structure of $R[S^{-1}]$, the localization of $R$ with respect ...
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Permutation calculator

I am studying the Mathieu group $M_{12}$ on the twelve letters $\infty,7,6,8,X,2,0,3,4,1,9,5$ (in this specific order) in the form that it is generated by the permutations $(0123456789X)$, ...
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1answer
55 views

Proof that $\sqrt{3} \notin \mathbb{Q}(\theta)$ where $\theta^4-2=0$. [on hold]

This is a problem in Robert Ash's lecture notes in Algebraic Number Theory. I have to prove that $\sqrt{3} \notin K=\mathbb{Q}(\theta)$ where $\theta^4-2=0$, using the fact that ...
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36 views

Can we always make a group? [duplicate]

Can we always find an operation on non-empty set, which create a group$?$ I cann't imagine, how not, but is it proof for that?
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6Sz as the automorphism group of the complex Leech lattice

Consider the Leech lattice as a complex lattice over the Eisenstein integers. Both in Conway ("Sphere packings, lattices and groups") and Wilson ("The Complex Leech Lattice and Maximal Subgroups of ...
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For a group epimorphism $f : G \to H$ with kernel $K$, prove that $G \simeq K \rtimes H$. Why is $G \simeq K\times H$ if $G$ is abelian? [on hold]

For a group epimorphism $f : G \to H$ with kernel $K$, prove that $G \simeq K \rtimes H$. Why is $G \simeq K\times H$ if $G$ is abelian? This question is from group theory in Abstract Algebra and ...
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3answers
440 views

Example of a ring with infinitely many zero divisors and finitely many invertible elements

I am preparing to my abstract algebra exam and I try to find an example of a ring with infinitely many zero divisors and finitely many invertible elements (rather simple if possible). Does it even ...
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5answers
801 views

Why is the commutator defined differently for groups and rings?

The commutator of two elements in a group is defined as $[g, h] = g^{−1}h^{−1}gh.$ In a ring, the commutator of two elements is $[a, b] = ab - ba.$ I'm asking because a ring is a (abelian) group ...
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2answers
32 views

Three polynomials as unknowns of an equation

If three polynomials $f,g,h\in\mathbb R[x]$ are such that $[f(x)]^2 –x[g(x)]^2+[h(x)]^2=0$, what can we conclude about $f, g, h$?
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1answer
18 views

classification of groups of order $4p, p\ge 5$, need help finding automorphism

So I've been working on this problem for my qual prep class, and I have it all down except for one detail. I'm doing it by semidirect products, and with the Sylow $p$ group normal, choosing the ...
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Projective general linear group on discrete valuation ring

Let $R$ be a complete discrete valuation ring and $k$ its residue field. Let $H$ be a finite subgroup of $PGL_2(k)$ such that its order is prime with char($k$). Is there some elementary way to show ...
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Covering groups

I am studying the Steiner system $S(5,6,12)$ and the ternary extended Golay code $\mathscr{C}_{12}$. The automorphism group of the Steiner system is the Mathieu group on twelve elements $M_{12}$ ...
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1answer
105 views

What motivates the definition of a ring in abstract algebra? [on hold]

I've read a lot of questions about rings but I still don't know why they are useful/ I'm sure they are, but I don't know why. Are their properties somehow used in proofs or as foundations for ...
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Valuations on $\Bbb Q(t)$

Ex. $2.3.3$ in Algebraic Number Theory by Neukirch is the following: Let $k$ be a field and $K = k(t)$ the function field in one variable. Show that the valuations $v_{\mathfrak p} $ associated to ...
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2answers
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There is no nontrivial ring homomorphism between two commutative rings with unity and characteristic of distinct primes

The following is an old exam question and the question is: Show that there is no nontrivial ring homomorphism between two commutative rings with identity if their characteristics are distinct primes. ...
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3answers
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How can I prove irreducibility of polynomial over a finite field?

I want to prove what $x^{10} +x^3+1$ is irreducible over a field $\mathbb F_{2}$ and $x^5$ + $x^4 +x^3 + x^2 +x -1$ is reducible over $\mathbb F_{3}$. As far as I know Eisenstein criteria won't ...
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1answer
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$M_1$, $M_2$ and $M_1\cap M_2$ injective imply $M_1+M_2$ is also injective?

Let $M$ be an $R$-module ($R$ is a ring with identity) and let $M_1$ and $M_2$ be two injective submodules such that $M_1\cap M_2$ is also injective. How to show $M_1+M_2$ is injective? If the ...
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1answer
52 views

Learning Galois theory - required subtopics that are prerequisite?

This is not a reference request, that is, I have access to many textbooks I am happy with. What I don't know is, what are the things I need to know to get started? My idea on the path of knowledge ...
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21 views

Fiber product of $f$ and $g$ is isomorphic to $\mathbb Z\oplus \mathbb Z_p$?

Let $p$ be a prime number. I'm supposed to show the fiber product (pullback) of the canonical projections $f:\mathbb Z\longrightarrow \mathbb Z_{p}$ and $g:\mathbb Z_{p^2}\longrightarrow \mathbb Z_p$ ...
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Proper formulation of one-to-one and onto proofs for group isomorphism

I have to construct an isomorphism for the two groups. I have the isomorphism itself but I'm not sure if my formulation is correct in regard to proving the mapping being 1-1 and onto and I don't want ...
8
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1answer
68 views

$A$ regular, $k'/k$ transcendental. How to prove that $A \otimes_k k'$ is regular?

Let $k$ be a field and $k'$ a purely transcendental extension of $k$. Let now $A$ be an integral finitely generated $k$-algebra. How to prove that if $A$ is regular then $A \otimes_k k'$ is also ...
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1answer
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Question about an inverse limit.

Define a partial order on $\Bbb{N}$ to be $n \leq m$ iff $n = m $ or $n |m$ and there's a twin prime dividing $m$ and not $n$. It's easy to see that it's a poset. Define a system of abelian groups ...
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The subring of $k[x, y]$ generated by $\{x y^{i}: i\geq 0\}$ is not finitely generated over $k$ [duplicate]

Let $R$ be the subring of $k[x, y]$ generated by $\{x y^{i}: i\geq 0\}$. Can someone explain why $R$ is not finitely generated as a ring over $k$ (i.e. finitely generated as a $k$-algebra)? By ...
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1answer
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Prime ideals in $R[x]$, $R$ a PID

Let $R$ be a PID. Show that if $r \in R$ and $$p = (r, \underline{f}(x), \underline{g}(x))$$ is prime, where $\underline{f}(x), \underline{g}(x) \in R[x]$ are nonconstant irreducible polynomials, ...
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Prime ideal $P$ of $\mathbb{Z}[x]$ such that $P \cap \mathbb{Z}=\{0\}$ is principal

The problem stated more precisely is this: Let $P$ be a prime ideal of $\mathbb{Z}[x]$ such that $P \cap \mathbb{Z} =\{0\}$. Show that $P$ is a principal ideal. I think there is a problem with my ...
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When a two-generated ideal of a noetherian integral domain have a finite projective resolution?

Let $R$ be a noetherian integral domain, and $I$ a non-zero ideal of $R$ which can be generated by two elements. (We do not know if $I$, considered as an $R$-module, is $R$-projective; maybe yes maybe ...
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1answer
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Conjugates of Sylow $p$-groups in $GL_3(F_p)$

In this list of review questions, there is the following question about $GL_3(F_p)$. Question 1.38. Let $G$ be the group of invertible 3 × 3 matrices over $F_p$, for $p$ prime. What does basic ...
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3answers
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Irreducible polynomial over $\mathbb{Q}(\zeta)$

Show that the polynomial $f(x)=x^5-2$ is irreducible over $\mathbb{Q}(\zeta)$, where $\zeta=e^{2\pi i/5}$. I tried show that the roots of polynomial $f(x)=x^5-2$, $$\sqrt[5]{2}, \zeta\sqrt[5]{2}, ...
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41 views

Projectivity of a (prime) ideal in a noetherian integral domain

Assume $R$ is a noetherian integral domain (and assume $R \neq k[x_1,\ldots,x_n]$), $I$ is a non-zero ideal of $R$ ($I$ is finitely generated, since $R$ is noetherian), and $I$ is not necessarily ...
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Prove $a_1a_2$ is commutative in ring $R$ if $a_1=a_2$ [on hold]

I was attempting to prove that any element of a ring squared is commutative in any given ring, and didn't know where to begin with this.
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2answers
64 views

Create a field from set of 2 elements.

Can we always create a field from a set of at least $2$ elements? For addition I considered a function: $A\times A \rightarrow A$. If $a+b=b+a=a+a \rightarrow a $. If $b+b \rightarrow b$. Is it ...
6
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1answer
115 views

A game from Exercise in Artin's Algebra (Chapter 2 M.13)

I found an interesting problem in Chapter 2 for Artin's Algebra (2nd Ed) in the Miscellaneous section that I haven't been able to figure out. The text of the problem is quoted below. M.13 (a ...
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3answers
58 views

Noncommutative algebraic operation. [on hold]

Can we always find a non-commutative algebraic operation in a non-empty set?
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1answer
28 views

Example of a Non-Graded Ideal in a Graded Ring

A ring $S$ is said to be graded if there are additive subgroups $S_0, S_1, S_2, \ldots$ such that $S=\bigoplus_{k\geq 0}S_k$ and $S_iS_j\subseteq S_{i+j}$ for all $i$ and $j$. An ideal $I$ in a ...