Abstract algebra is the study of algebraic objects. Some of the more common algebraic objects are groups, rings, fields, vector spaces, modules, and other advanced topics.

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Hochschild cohomology of skew polynomial rings

Definitions The skew polynomial algebra over $\mathbb{C}$ is defined as $\mathbb{C}<x,y>/(xy-yx+x)$ or alternatively as $\mathbb{C}[x,y,\sigma]$, where $\sigma$ is the automorphism on ...
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Ring homomorphism between polynomial rings and its kernel

Suppose $G$ is a finite group and $x\in \mathbb{C}$, $u\in G$ are elements of order $p$, so $x^p = u^p = 1$. Consider the ring homomorphism $\varphi: \mathbb{Z} [u] \rightarrow \mathbb{Z}[x]$ defined ...
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Evaluate this product $n \times \frac{n-1}{2} \times \dots \times \frac{n-(2^i-1)}{2^i}$

For $i = \lfloor \log_{2}(n+1) \rfloor - 1$ evaluate $n \times \frac{n-1}{2} \times\frac{n-3}{4} \times \frac{n-7}{8} \times \dots \times \frac{n-(2^{i}-1)}{2^i}$ So the product goes up to $i$ and I ...
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Quotient of polynomials, PID but not Euclidean domain?

While trying to look up examples of PIDs that are not Euclidean domains, I found a statement (without reference) on the 'Euclidean domains' page of Wikipedia that $$\mathbb{R}[x,y]/(x^2+y^2+1)$$ is ...
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1answer
39 views

Star in Serre duality

Why is there a dual bundle in Serre duality? Let $\mathcal E$ be a vector bundle over complex manifold $X$, without any metric anywhere, then one has a pairing $$(\Omega^{0,q} \otimes \mathcal E) ...
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A question on the dual relationship between the regressive product and the exterior product

I am trying to understand the following sentence, which I came across in a book: The underlying beauty of the Ausdehnungslehre is due to this symmetry [the duality between the regressive and ...
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1answer
27 views

The minimum size of generating set of the external direct product

I have seen the following theorem here: Suppose that $A$ and $B$ are finite groups whose orders are relatively prime to each other, and the minimum size of generating set (i.e., the smallest ...
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25 views

Are most univariate pyramidal polynomials over $\mathbb{Z}$ or $\mathbb{Z}/p$ reducible or irreducible?

Consider the special family of polynomials of odd degree where the following condition holds : $\forall i$, $a_i > 0$ and $a_d < a_{d-1} < \ldots a_{d/2+1} > a_{d/2} > ... > a_0$ ...
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1answer
48 views

How many elements of order 4 does $S_6$ have?

I am trying to count the number of elements of order 4 in $S_6,$ but my answer is not matching the one in the back of the book. Here's my attempt: Such elements are either of the form $(6543)(21)$ ...
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1answer
64 views

Radical Solvable quintic polynomial.

I am trying to solve the following exercise: Let $k$ be a field of characteristic 0, and let $f(x)\in k[x]$ be a polynomial of degree 5 with splitting field $E/k$. Prove that $f(x)$ is ...
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a second course in abstract algebra

I recently read an abstract algebra textbook, "A first course in abstract algebra" by John Fraleigh. I am interested in continuing to do some more self studying. What is a good book for a second ...
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59 views

alternate proof of the fundamental theorem of algebra

I was reading over my notes from complex analysis and saw the fundamental theorem of algebra which states that: A polynomial of positive degree over a field $\mathbb{C}$ of complex numbers has a ...
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1answer
24 views

Basis for the completion of a free module

This (or similar) question might have been asked before- apologies for any duplication. I've got a Dedekind domain $R$, a non-zero prime ideal $P$ of $R$ and the completion $\widehat{R}$ of $R$ wrt ...
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1answer
48 views

Localization and direct limit

Let $ A $ be a ring. Let $ I $ be a preordered set, filtering. Let $ \Sigma $ a multiplicative subset of $ A $. Suppose for any given $ i \in I $ a multiplicative subset $ S_i $ of $ A $ contained ...
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24 views

Show that $I(a,b)=I(a',b')$

Please help me to solve this problem: "Let $a,b,a',b',m,n,r,s$ be integers such that $m.s-n.r=1$ or $m.s-n.r=-1$, $a'=m.a+n.b$ and $b´=r.a+s.b$. Show that $I(a,b)=I(a',b')$, where $I$ is the symbol ...
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How to find a minimal polynomial

I want to solve the following problem. But I have no idea. I can't come up with what I have to do first. Help me please. Put $L=\mathbb{C}(X,Y,Z)$, $\omega=\frac{-1+\sqrt{-3}}{2}$. Define two ...
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1answer
34 views

If $A\ne 0$ is a square matrix over a commutative ring with $\det A=0$, then its null space contains an element whose components are minors of $A$

Let $R$ denote a commutative ring and $A\ne 0$ a $n\times n$ matrix over $R$ with $\det A=0$. Then there exists a $x\in\ker A\setminus\left\{0\right\}$ such that all components of $x$ are minors of ...
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1answer
39 views

$(u+1)(u+2)\cdots(u+p)=u^p-u$

Let $E$ be a field with characteristic $p>0$. How should I prove that $$(u+1)(u+2)\cdots(u+p)=u^p-u$$ I can verify this equality for small $p=2,3,5$. Is there a way to prove this result in general? ...
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35 views

Ring theory(addition table)

(S,+,.) is a ring , where S={a,b,c,d}. Complete the table. $$ \begin{array}{c|ccccc} + & a & b & c & d \\ \hline a &a &b &c &d \\ b &b &1 &2 &3\\ ...
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What is the meaning of “fix” in field theory?

What is the meaning of "fix" in field theory? Example: I found a definition of field automorphism, A field automorphism fixes the smallest field containing $1$, which is $\Bbb Q$, the rational ...
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zeros of a simple polynomial

For $x, y \in GF[2^n]$, consider the two-parameter polynomial $P(x,y) = x \cdot y + f(x) + g(y)$, where $f$ and $g$ are arbitrary polynomials on $GF[2^n]$. Can we say anything about the number of ...
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Does there exist two non-constant polynomials $f(x),g(x)\in\mathbb Z[x]$ such that for all integers $m,n$, gcd$(f(m),g(n))=1$?

Does there exist two non-constant polynomials $f(x),g(x)\in\mathbb Z[x]$ such that for all integers $m,n$, gcd$(f(m),g(n))=1$? I think there are no such polynomials, but how to prove?
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A question about separable extension

We say $K$ is separably generated over $k$ if there exists a transcendence basis $\{x_i,i\in I\}$ of $K/k$ such that $K/k(x_i,i\in I)$ is a separable algebraic extension. We say $K$ is separable over ...
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49 views

Prove that $k(\alpha+\beta)=k(\alpha,\beta)$

I am trying to solve the following problem: Let $k$ be a finite field and let $k(\alpha,\beta)/k$ be finite. If $k(\alpha)\cap k(\beta)=k$, prove that $k(\alpha,\beta)=k(\alpha+\beta)$. What I ...
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1answer
38 views

Intuitive explanation for a map $z \to z^p$.

Let $p$ be prime and let $G$ be the group of $p$-power roots of 1 in $\mathbb{C}$ . Prove the map $z \to z^p$ is a surjective homomorphism. $\textbf{My Attempt:}$ $G=\{ z \in \mathbb{C} \mid ...
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2answers
28 views

Composition of ring homomorphism

I have three rings $A,B,C$ and ring homomorphisms $f: A \rightarrow B$ and $g: B \rightarrow C$, which are both surjective. Is it true that $C$ is isomorphic to $$ A / (\ker(f), \ker(g)) ? $$ If so ...
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1answer
62 views

How to prove this innocent looking isomorphism

I've got a Dedekind domain $R$ with quotient field $K$, a non-zero prime ideal $P$ of $R$. I form the completion $\widehat{K}$ of $K$ wrt the valuation $v_P$ associated to $P$. Let $\widehat{R}$ be ...
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Does every group act faithfully on some group?

Cayley's Theorem shows that every group acts faithfully on some set. In other words, one can find an injective group homomorphism $\sigma: G\to S_{A}$ where $S$ is the set of all bijections on some ...
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Finite-dimensional algebras with invertible elements and without idempotents

Does there exist a finite-dimensional algebra containing a nontrivial invertible element, with no nontrivial idempotents? By 'nontrivial', I mean not proportional to the identity.
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2answers
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If a group has no maximal subgroups then all elements are non-generators? Frattini subgroup characterization

This question is the last leg of an exercise I've been working on in which we characterize the intersection of all maximal subgroups as the subgroup of all non-generators. I've already shown that if a ...
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2answers
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Jacobson radical in $A[x]$ where $A$ is a ring.

Let $A$ be a commutative ring, and $A[x]$ be the ring of polynomials in an indeterminate x, with coefficients in A. I found several proofs online that in this case the Jacobson radical equals to the ...
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1answer
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calculating signature and showing group homomorphism

I got this question in my group theory book. I think I understand the theory behind it but can't seem to use it to get a solution im happy with. Let $V = M_2(\mathbb F)$. For $x,y \in V$ define ...
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Let $\mathbb Q\le K$ finite extension of fields $\alpha\in K$. Then $N(\alpha)=\pm1\Leftrightarrow\alpha$ is a unit

Let $\mathbb Q\le K$ (where $K\le\mathbb C$) be a finite extension (say $|K:\mathbb Q|=n$) of fields and let $\alpha\in K$ be an algebraic integer, i.e. $\alpha$ is a root of a monic polynomial over ...
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1answer
37 views

Show that $(x_1\times\dots\times x_k)\times x_{k+1}=x_1\times\dots\times x_{k+1}$

Let $a\times b\times c$ denote $a\times(b\times c)$. Given $(a\times b)\times c=a\times(b\times c)$, how do you prove $$(x_1\times\dots\times x_k)\times x_{k+1}=x_1\times\dots\times x_{k+1}$$ ? ...
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Order of the elements of a right coset [on hold]

What can we say about order of the elements of a right coset in a finite group.
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Linear representation of a free group

I need to prove the following: "Prove that the free group of rank 2 is linear." So to my best understanding, and please correct me if I'm wrong: I actually need to show a homomorphism from the free ...
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126 views

A question about Sylow subgroups

Let $G$ be a finite group and $P\neq\{e\}$ be a Sylow $p$-subgroup of $G$ and $P^g\neq P$ be its conjugate in $G$. If we know that $P\cap P^g\neq \{e\}$, can we conclude that $Z(P)\cap Z(P^g)\neq ...
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2answers
49 views

Ring Structure: Definition

Let $R = \{a+bi\mid a,b \in \mathbb Z, i^2=-1\}$, with addition and multiplication defined by $(a+bi)+(c+di)=(a+c)+(b+d)i$ and $(a+bi)(c+di)=(ac-bd)+(bc+ad)i$, respectively. (a) Verify that $R$ is ...
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Finite generated abelian group $G$ and $H<G$. What is the rank of $(G/H)/(G/H)_t$?

I saw another question about this problem here. However there are quite different answers from my expectation. Anyway, here are my trials. Trial 1 : By structure theorem, $G\cong G_t\oplus F_1$ ...
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About product order [on hold]

Are there any references talking about product order on this wikipeida link? thanks!
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Linear algebraic group

Let A be a finite dimensional algebra over C . This means that there is a multiplication map $f : A \times A \to A$ that is bilinear ( it is not assumed to be associative). Define the automorphism ...
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Proving that finite direct and inverse limits exist in an additive category having kernels and cokernels

I have attempted to prove this but am unable to complete the proof. Below is my attempt. Let $\mathcal{A}$ be a category satisfying the conditions in the title and $\{M_i,\phi^i_j\}$ be a finite ...
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Units of $\mathbb{Z}[\sqrt[4]2]$

How would one compute the units in $\mathbb{Z}[\sqrt[4]2]$? According to one source, it can be shown that the fundamental units are $1 + \sqrt[4]2$ and $1 + \sqrt{2}$, but it does not specify the ...
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Can I assign a distinct homomorphism to every function?

I consider an algebraic structure and particular structure preserving morphisms (think groups and group homomorphism). I wonder if I can assign a distinct such morphisms to every function between any ...
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44 views

can somebody provide me this paper by NIVAN? [on hold]

equations in quaternions by I. Nivan published in AMM Vol 48 , 654-661 (1941). please i need it, and could not find it online anywhere.
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Conjugates and normalizer

Let $H=\langle(1 2 3)\rangle$ and let $G=S_3$. Now, $$(1 2)(1 2 3)(1 2)=(1 3 2)=(1 2 3)^{-1}$$ Since $(1 2)$ conjugates a generator of $H$ to another generator of $H$, we can conclude that $(1 2) ...
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2answers
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Show that $A = \{(3x,y)~|~ x,y \in Z\}$ is a maximal ideal of $Z \oplus Z$. My Attempt Shown

Show that $A = \{(3x,y)~|~ x,y \in Z\}$ is a maximal ideal of $Z \oplus Z$. Here's my attempt, Please tell me where did I go wrong. Attempt: When $R$ is a commutative ring with unity and $I$ is any ...
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33 views

$p$-completion of a $\mathbb{Z}_p$-module

Let $p$ be a prime number, $\mathbb Z_p$ the ring of $p$-adic integers. Let $M$ be a finitely generated $\mathbb Z_p$-module and $\widehat{M}$ its $p$-completion $\varprojlim_n M/{p^n M}$. ...
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28 views

Must the index $k=|G:HC_G(x)|$ be finite?

I want to solve the following Exercise from Dummit & Foote's Abstract Algebra text: Assume $H$ is a normal subgroup of $G$, $\mathcal{K}$ is a conjugacy class of $G$ contained in $H$ and $x ...
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1answer
31 views

Action of GL$(2,\mathbb{R})$ on symmetric matrices

This is a problem from an old qualifier. Let GL$(2,\mathbb{R})$ act on SYM, the real symmetric 2x2 matrices, via $S \mapsto A^T SA$ for $A \in$ GL$(2,\mathbb{R})$ and $S \in$SYM. Show that each ...