Questions tagged [abstract-algebra]
For questions about monoids, groups, rings, modules, fields, vector spaces, algebras over fields, various types of lattices, and other such algebraic objects. Associate with related tags like [group-theory], [ring-theory], [modules], etc. as necessary to clarify which topic of abstract algebra is most related to your question and help other users when searching.
81,528
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Where Does This Proof Use $\text{Char}(K)=p$?
$\DeclareMathOperator{\char}{char}$
$\DeclareMathOperator{\aut}{Aut}$
I've been reading over the following theorem (5.7.7) from Hungerford's algebra, and two things are confusing me about the proof of ...
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use of binary operation to represent hadamard transform
In very frank terms, meaning basic binary operations and well-defined definitions, can someone explain to me what this operation is supposed to mean and what the notation is supposed to represent? I ...
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How can you have an isometry without a metric/coordinates?
I'm self-studying Artin's Algebra and I'm getting a bit thrown through a loop on isometries (in Chapter 6, in case you're curious). Specifically, a section about change of coordinates:
Let $P$ denote ...
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Proving the Relative Nullstellensatz
Let $X \subseteq \mathbb{A}^n$ be an affine subvariety and let $A(X)$ be the corresponding coordinate ring. For $Y \subseteq X$ and $S \subseteq A(X)$ we define the following relative analogues to the ...
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What is the point when the profinite topology of a group $G$ induces the full profinite topology on a subgroup?
I have a question. I don't know if is trivial or even it makes sense, but is something I could not understand. For example, consider the following exercise of Ribes-Zalesskii:
Exercise 9.1.1 Let $G = ...
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Why does the semigroup of matrices form an epigroup?
An epigroup is a semigroup $S$ in which every element has a (positive) power that lies in a subgroup of $S$. (The subgroup may depend on the element). Note that if $x^n\in G$, where $G$ is a subgroup ...
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Question on proof that all subgroups of $\mathbb{Z}$ and $\mathbb{Z_n}$ are subrings (and ideals).
I have 2 related problems: Every subgroup of $\mathbb{Z}$ (under addition) is a subring (and an ideal), and the same thing but for $\mathbb{Z_n}$ (I'm following the definition for rings that doesn't ...
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If the cardinality of the quotient set of a subgroup is $1$ does that mean that the subgroup is equal to the whole group?
Let $(G, \cdot)$ be a group and let $H \leq G$ (i.e. $H$ is a subgroup of $G$) such that $\mid G:H \mid = 1$ (i.e. the index of $H$ in $G$ is $1$). Does this imply that $G=H$?
In the case that $G$ is ...
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The Coordinate ring $A(X)$ is a field if and only if $X$ is a singleton
Let $X \subseteq \mathbb{A}^n$ be an affine variety. Show that the coordinate ring $A(X)$ is a field if and only if $X$ is a singleton.
Remark: This is exercise 1.20 from Gathmann's Notes "...
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Minimal polynomial of $\mathbb C/\mathbb R$
We know that the extension $\mathbb C/\mathbb R$ is finite and finitely generated with degree 2, as $\left\lbrace1,i\right\rbrace$ is a basis.
How could we find the minimal polynomial (degree 2) of ...
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kernel of subgroup homomorphism
Let $f:A \longrightarrow B$ be a group homomorphism, and note $C$ a subgroup of $A$ and $D$ a subgroup of $B$. Can we find a link between the kernel of $f$ and the kernel of the group homomorphism $g: ...
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Show that the image of an irreducible affine variety is irreducible [duplicate]
I would really appreciate some help on this:
Definition: Let $k$ be a field. An affine variety is a space with functions $(X, O_X)$ i.e. a topological space equipped with a sheaf of $k-algebras$, ...
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Why doesn't $x^4+14x^2+49$ reducible over $C$?
I was following James Cook's Abstract Algebra at the moment, in his video https://www.youtube.com/watch?v=uo8ogufaSKA&list=PLBY4G2o7DhF0JCgapYKrqibGaJuvV4Gkb&index=41 at $36:26$ he said that $...
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Number of involutions in $S_n$? [duplicate]
I was having some fun with the number of involutions (didn't know they were called that) in the symmetric group $\Psi(S_n$), and tried to come up with a simple formula for it, I'm like, so close to ...
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Regular module is Indecomposable
Suppose that $R$ is a PID. How would I go about showing that $R$ as a left $R$-module is indecomposable. That is, it cannot be written as a direct sum of two non-zero submodules.
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Proof verification: $[L_1L_2:K] \leq [L_1:K][L_2:K]$
Let $K \subseteq L_1,L_2 \subseteq L$ be fields. I want to show the following:
$$[L_1L_2:K] \leq [L_1:K][L_2:K]$$
My attempt:
The definition of $$L_1L_2:=L_1(L_2):=\bigcap \{F:F\text{ is a subfield ...
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Examples of subgroups of additive group that aren't subrings
I just started learning about rings, and when playing around with some examples I noticed that in all of them, every subgroup of their additive group was also a subring.
So, I'm looking for examples ...
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Let $F$ be a field; what does $(F)$ mean?
Let $L,K,L_1,L_2$ be fields where $L/K$ is a field extension and $K \leq L_1,L_2 \leq L$ be intermediate fields.
We define $L_1L_2$ to be the subfield of $L$ generated by $L_1 \cup L_2$
Show the ...
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$\mathbb{C}$ is complete
Can
$\mathbb{R}^2 \simeq \mathbb{C}$ and because $\mathbb{R}^2$ is complete $\implies \mathbb{C} $ is complete
be an argument to show thaf $\mathbb{C}$ is complete? Or can you give me a proof please?...
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Monoid with torsion elements
I am currently studying construction of the Grothendieck group of a commutative monoid $M$. I was looking for an example of a monoid that is torsion, namely, I have the following query.
Does there ...
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how to find the invariant divisor of a quotient module? [closed]
let $D=\mathbb{Z}[\sqrt{-1}]$, $K$ is the submodule of $D^3$ generated by$f_1=(1,3,6), f_2=(2+3\sqrt{-1}, -3\sqrt{-1}, 12-18\sqrt{-1}), f_3=(2-3\sqrt{-1}, 6+9\sqrt{-1}, -18\sqrt{-1}).$
then how to ...
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projective and injective ideals of $\mathbb Z/n\mathbb Z$ as $\mathbb Z/n\mathbb Z$ module
Question:
I want to determine when the ideals of $\mathbb{Z/nZ}$ as $\mathbb{Z/nZ}$ module are projective/injective modules.
Here are some of the cases:
Proj:
(0)$\mathbb{Z/nZ}$ itself is projective $\...
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How do i show that $(1, 4)\sigma (1, 4)^{−1}$ is not an element of $\langle \sigma \rangle$?
I have
$$\sigma = \binom{1\ 2\ 3\ 4\ 5}{2\ 5\ 4\ 3\ 1} \in S_5$$
How do I show that $(1, 4)\sigma (1, 4)^{−1}$ is not an element of $\langle \sigma \rangle$?
(I don't know even what $(1, 4)\sigma (1,...
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Mellin transform defined for function on group $(\Bbb R^+,\times)$ but integration domain is over semigroup
I came across this question: Mellin transform defined for function on group $(\mathbb{R}^{+}, \times)$.
Let $\varphi_x: s\mapsto x^s $ be a group isomorphism from $(\Bbb R,+)$ to $(\Bbb R^+,\times).$
...
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Proof: $S = \{[2m]_{1000} : m = 0, 1, ... , 500\}$ is an Ideal of $R = \mathbb{Z}_{1000}$
It is given in the question that $S \subset R$, so it is not necessary to prove this, in my understanding I only need to show that $S$ is closed under addition and multiplication with elements from $R$...
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Is every ring homomorphism between real algebras also real-linear?
$\def\bbR{\mathbb{R}}
\def\bbQ{\mathbb{Q}}$The comment from Vladimir Sotirov in March 2022 in this answer could be interpreted as suggesting the possibility that every ring homomorphism between $\bbR$-...
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Does division by 0 really have anything to do with $\lim_{x \to 0} \frac{1}{x}$?
To my very limited knowledge, division by 0 is undefined precisely because it breaks the field axioms. No dividing by 0 if you want a field. However there do exist structures that are not fields which ...
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understanding algebraic proof in complex geometry
I am studying "Condensed Mathematics and Complex Geometry" by Dustin Clausen, Peter Scholze. I came across this theorem and this proof:
I don't understand a lot of steps in the proof. I ...
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Prove every Sylow Subgroup of $G$ is Cyclic
I'm working on the following problem. Let $G$ be a finite group such that for each $n \mid |G|$, we have that $G$ contains at most $1$ subgroup of order $n$. Prove that every Sylow $p$ subgroup of $...
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Proof verification: Herstein's Abstract Algebra, Problem 28 from Sec 4.5 [duplicate]
I will paraphrase the problem since it refers to a previous problem from the section:
Let $R$ be a commutative ring, $q(x)$ a zero divisor in the ring $R[x]$. If
$$ q(x) = a_0 + a_1x + \cdots + a_nx^...
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If $N$ is a subset of a monoid $(M,\bot ,e)$ with identity element $\epsilon$ with respect $\bot$ then does the equality $\epsilon=e$ holds?
If $(M,\bot,e)$ is a monoid then it is usually to say that $N$ in $\mathcal P(M)$ is a submonoid of $(M,\bot, e)$ if it is closed under $\bot$ and it contains $e$. So by this definition I suspect that ...
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Finding a representation of the inverse as linear combination
I am struggling with the following:
Write $(7+2^{\frac{1}{3}})^{-1} \in \mathbb{Q}(2^{\frac{1}{3}})$ as a $\mathbb{Q}$-Linear combination of $\{1,2^{\frac{1}{3}},2^{\frac{2}{3}}\}$. Hint: This means ...
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Find the possible values for $[\mathbb{Q}(\alpha,\beta):\mathbb{Q}]$ and provide an example of $(\alpha,\beta)$ for each possible value
Let $\alpha, \beta \in \mathbb{C}$ s.t. $[\mathbb{Q}(\alpha):\mathbb{Q}] = [\mathbb{Q}(\beta):\mathbb{Q}] = 4$
Find the possible values for $[\mathbb{Q}(\alpha,\beta):\mathbb{Q}]$ and provide an ...
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Is it possible to recover the Cartan-Leray Spectral Sequence for Group Cohomology from the Grothendieck Spectral Sequence?
Let $G$ be a discrete group acting freely and cellularily on a CW-complex $X$.
I am interested in the Cartan-Leray spectral sequence from Eilenberg and Cartan's Homological Algebra, Theorem XVI.8.4, ...
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show that Ĝ is non empty and an abelian group when equipped with pointwise operations. [closed]
Assuming G is a locally compact group ( not abelian). A character of G is a continuous group homomorphism x: G → 𝕋 , where 𝕋 is the circle group . Denote by Ĝ the set of characters of G.
Here how ...
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Dense algebraic curves
Let's assume we are given $\mathbb{R}^2$ with the standard topology (unions of open balls).
We call a subset $A \subset \mathbb{R}^2$ dense, if $\forall U \underset{open}{\subset} \mathbb{R}^2 \ \...
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Definition of Presentation of Group in Dummit’s Abstract Algebra [closed]
A subset $S$ of elements of a group $G$ with the property that every element of $G$ can be written as a (finite) product of elements of $S$ and their inverses is called a set of generators of $G$. We ...
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Characteristic of a quotient ring in $\mathbb Z[i]$ [duplicate]
I would like to solve this exercise:
Let $\mathbb Z[i]$ be the Gaussian integers ring.
Check whether the ideal $I$ generated by $2−i$ is prime in $\mathbb Z[i]$.
Find the characteristic of quotient ...
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Lifting Primary Decomposition Over DVRs From Their Field of Fractions
Suppose we have an ideal $I = (f_1,...,f_m) \subset \mathcal{O}[x_1,...,x_n]$ where $\mathcal{O}$ is a DVR with uniformizer $r$. Let $K := \text{Frac}(\mathcal{O})$. Let $I_K$ be the ideal of $K[x_1,.....
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polynomial with algebraic coefficient
Let $K/F$ be a field extension and f a polynomial in $F[x]$. If $g$ divides $f$ in $K[x]$ and $g$ is monic, we have to show that the coefficients of $g$ are algebraic over $F$.
This is a question I am ...
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Prove that $\sqrt{-5}$ is a prime in the ring $R=ℤ[\sqrt{-5}]$.
If $R=ℤ[\sqrt{-5}]$ is a ring but not a UFD, prove that the irreducible element $\sqrt{-5}$ is a prime.
This is what I have so far.
Proof: Let $R=ℤ[\sqrt{-5}]$ be a ring but not a UFD. Since $\sqrt{-5}...
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How to write Type III matrix as a product of Type I matrices?
I saw this question Prove that, over a Euclidean domain $R, I + cE_{ij}$ generate $SL_n(R)$. and in the comment section, someone mentioned that you can do something about type III matrices.
I tried to ...
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Zero divisors in $\mathbb Q [T]/(T^4+T^2+1)$
I am asked to find all zero divisors in the quotient ring $\mathbb Q [T]/(T^4+T^2+1)$.
The most that I have come to is to take into account that if that set is a field and the polynomial is ...
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If Galois group is $D_8$ then show that an intermediate extension has quadratic subfield
Given that $F$ is degree extension $4$ of $\mathbb Q$ which is not Galois. If $K$ is Galois closure of $F$ and $\operatorname{Gal}(K/\mathbb Q)$ is $D_8$ then I have to show that $F$ has subfield that ...
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Algebraic structures with opposite subalgebra lattices and congruence lattices that are not groups with operators
Are there any algebraic structures where the congruence lattice and subalgebra lattice are opposites that do not look like groups with operators?
My motivation for this question is noticing that ...
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Noetherian rings and greatest common divisors of ideals.
I'm reading this Algebra book at the moment and found an exercise that I've been struggling with for quite a while now:
Let $R$ be a ring and $M$ be the smallest set containing all principal ideals of ...
- 1
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1
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25
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Describe an ideal in a residual polynomial ring
Given the polynomial ring over a field $\mathbb{K}[x]$ and $f\in \mathbb{K}[x],$ one may define the residual polynomial ring $\mathbb{K}[x]/(f).$ I am wondering how does an ideal of the residual ring ...
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Which general methods are there for determining if $t$ and $e^t$, where $t$ is any given transcendental number, are algebraically independent?
Which general methods are there for determining if $t$ and $e^t$, where $t$ is any given transcendental number, are algebraically independent?
Can Schanuel's conjecture be used for this?
We have ...
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Let $F$ be a field. Every (non-constant) polynomial $p \in F[X]$ is transcendental over $F$
I will start with the Definition:
Let $L/K$ be a Field extension. The element $a \in L$ is called algebraic over $K$ if there exists a polynomial $p \in K[X]$ such that $p \neq 0$ and $p(a)=0$.
If ...
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If $B$ is an integrally closed domain and $B \to A$ is an integral extension, then $A$ is the integral closure of $B$ in $K(A)$ or not?
Here $K(A)$ and $K(B)$ mean the fraction fields of $A$ and $B$, respectively. I see that the integral closure of $B$ in $K(A)$, denoting it as $C$, satisfies $A \subseteq C$ for $B\to A$ being an ...
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