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.
87,884 questions
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If $K/F$ and $L/F$ are separable, then the composite $KL/F$ is separable
Prove that, if $K/F$ and $L/F$ are separable, then the composite $KL/F$ is separable.
My attempt: for finite extensions, this useful lemma holds:
Let $K/F$ be a field extension. If $\alpha_1, \dots, ...
- 1,177
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Classification of Cyclotomic Skewfields?
I say that a skewfield (division ring) $D$ over $\mathbb{Q}$ is cyclotomic whenever $D$ admits a finite $\mathbb{Q}$-basis $\{\zeta_1,\dots,\zeta_n\}$ with each $\zeta_i$ a root of unity (i.e., $\...
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Do division rings have algebraic closures?
The theorem of existence of the algebraic closure of a field can be written as the following assertion:
Let $Q$ be a field, then there exists a field $C$ (having the same characteristic than $Q$) ...
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If $f(x)$ is irreducible over $\mathbb{Q}$, the equivalence classes of roots of $f(x)$ under transpositions of the Galois group are the same size
Let $f(x) \in \mathbb{Q}[x]$ be an irreducible polynomial over $\mathbb{Q}$ and let $R$ be set of roots of $f(x)$ in a splitting field $K$ (over $\mathbb{Q}$). Call $G$ the Galois group of $K/\mathbb{...
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Linear factors of polynomial in a polynomial ring
What is the fastest known method to split $f \in \mathbb{Z}[x]$ into linear factors mod $g \in \mathbb{Z}[y]$, assuming this happens?
Since that is not very clear, here is an example of what I'm ...
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Prove that the extension $\mathbb{Q} \subset \mathbb{Q}(\sqrt{3 + \sqrt{7}})$ is not normal
I want to prove that the extension $\mathbb{Q} \subset \mathbb{Q}(\sqrt{3 + \sqrt{7}}) = L$ is not normal. My strategy is to show that the minimal polynomial $f(x) = x^4 - 6x^2 + 2$ of $\alpha = \sqrt{...
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Chinese remainder theorem in rngs (rings without identity)
Hungerford states the Chinese remainder theorem as follows.
Let $A_1,\cdots,A_n$ be ideals in a ring R such that $$\begin{cases} R^2 + A_i = R\ \ \ \forall i \\ A_i+A_j = R\ \ \ \ \forall i \ne j \end{...
- 2,611
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Main theorem of Galois Theory: professor Borcherds' counting argument
In this lecture, professor Borcherds gives a great proof of the main theorem of Galois Theory. My question is about the counting argument he uses to prove the following fact.
If $M/K$ is a Galois ...
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What is an example of a finite, indecomposable semigroup with left identity and right inverses other than the basic example $g \ast h := h$?
This old question asks for an example of a semigroup $(G, \ast)$ for which
(a) $G$ has a left identity, i.e., an element $e \in G$ such that $e \ast g = g$ for all $g$, and
(b) every element of $G$ ...
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Must the spectral map $\operatorname{Spec}B\to\operatorname{Spec}A$ be surjective if the image contains all closed points of $\operatorname{Spec}A$?
Let $A\to B$ be an injective homomorphism of commutative rings. If the image of the spectral map $f:\operatorname{Spec}(B)\to\operatorname{Spec}(A)$ contains all closed points of $\operatorname{Spec}(...
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Prove that Q/Z is isomorphic to a subgroup of SO2(R) [duplicate]
I get that we need to find a homomorphism from Q/Z to SO2(R) with kernel Z and then apply the first isomorphism theorem. However I got stuck at defining this homomorphism.
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If $K/L$ is normal and $L/F$ is purely inseparable, then $K/F$ is normal - How to finish the proof?
This question is related to If $F/L$ is normal and $L/K$ is purely inseparable, then $F/K$ is normal .
Let $K/L$ and $L/F$ be field extensions. if $K/L$ is normal and $L/F$ is purely inseparable, ...
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Prove $x^{46}+69x+2025$ is irreducible in $\mathbb Z[x]$
I was told to work in $\mathbb F_{23}$, and also show it has a linear factor $\mathbb Z_5$
Write $f(x)=x^{46}+69x+2025$. We begin by supposing that $f=gh$ for some $g,h \in \mathbb Z[x]$. First, $g$ ...
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How to show that $\operatorname{lim}_{n \in \mathbb{N}} R[x_0, \ldots, x_n] \cong \operatorname{lim}_{(A, p(t))} A$?
Let $R$ be a commutative ring with unity. Let $I$ be the following index category:
The objects of $I$ are pairs $(A, p(t))$ where $A$ is an $R$-algebra and $p(t) \in A[t]$ is a polynomial
A morphism ...
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Associated polynomial function vs evaluation map
In Bosch's "Algebra: From the Viewpoint of Galois Theory" (page 29), the author considers a ring extension $R\subset R'$, a polynomial $f=\sum_ia_iX^i\in R[X]$, and says that we can ...
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Example of ring with an element that is a left zero divisor, but not a right zero divisor
I understand that in a commutative ring (with unity) all left zero divisors are also right zero divisors. Do we just talk about zero divisors.
From Wikipedia I see that this is not the case if $R$ is ...
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Compute multiplication of power of the irrelevant ideal of the associated graded ring with the associated graded module.
Brief question. Let $(R,\mathfrak{m})$ be a Noetherian local ring with $\mathfrak{q} \subseteq R$ an ideal. Let $M$ be a finitely generated $R$-module. Then for $n \ge 0$ nonnegative integer,
$$ (( \...
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Valuation domain with principal maximal ideal
Let $V$ be a valuation domain whose maximal ideal $M = (t)$ is principal.
I want to prove that
$\bigcap_{n \geq 1} (t^n) = (0)$
Suppose that $0 \neq x \in \bigcap_{n \geq 1} (t^n)$.
Then for each $n$...
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Complement direct summand to a non-principal ideal in a number ring
Let $K$ be a number field, that is, a finite extension of ${\mathbb Q}$. Let $R={\mathcal O}_K$ be the Dedekind domain of algebraic integers in $K$.
It is well-known that an ideal $I\subset R$ is a ...
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Coinvariants and finite index subgroups
Let $M$ be a $G$-module for some group $G$, and let $H$ be a finite index subgroup of $G$.
Let $M_G$ denote the coinvariants of $G$ acting on $M$. So $M_G = M / K$ where $K < M$ is the subspace ...
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3
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173
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How to show that the natural map $\mathbb{Z} \to \mathbb{Z}[1/n] \times \mathbb{Z}/\langle n \rangle$ is an epimorphism?
How can we show that the natural map $\newcommand\quotient[2]{{^{\Large #1}}/{_{\Large #2}}} \mathbb{Z} \to \mathbb{Z} \left[ \frac{1}{n} \right] \times \quotient{\mathbb{Z}}{\langle n \rangle}$ is an ...
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Equivalence of categories between quiver representations and module for infinite quiver
Every time I look for resources, authors assume quivers to be finite. I’m sure this question has been answered somewhere, but I cannot find it. I am reading Assem’s book on the representation theory ...
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If $\varphi: A \to B$ is an epimorphism and $J \subseteq B$ is an ideal with $f^{-1}(J) = \{0\}$, does it follow that $J = \{0\}$?
Let $R$ be a commutative ring with unity. Let $A$ and $B$ be commutative $R$-algebras. Let $\varphi: A \to B$ be an $R$-algebra homomorphism which is an epimorphism. Let $J \subseteq B$ be an ideal. ...
- 4,822
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Matrices over a PID
I was working on some proofs of slight variants of the Smith canonical form on square matrices over a PID $R$.
In particular, the context is the following: let $R=K[t,t^{-1}]$ where $K$ is an ...
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If $A \to B$ is universal with respect to sending $I$ to the unit ideal and $A$ is an integral domain, does $B$ also have to be an integral domain?
Let $R$ be a commutative ring with unity. Let $A$ be a commutative $R$-algebra. Let $I \subseteq A$ be an ideal with $I \neq \{0\}$. Let $\varphi: A \to B$ be a map of $R$-algebras satisfying the ...
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