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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Projections $P$ and $Q$ such that $I-(P+Q)$ is invertible.

Let $P,Q$ be endomorphisms of a finite dimensional linear space, such that $P^2 = P$ and $Q^2 = Q$. If $I-(P+Q)$ is invertible, then $P$ and $Q$ has the same rank. The solution is that $rk(P) = ...
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A normal subgroup problem

Let $G$ be a group in which, for some integer $n>1$, $(ab)^{n}=a^{n}b^{n}$ for all $a,b \in G$. Show that $G^{(n)}=\{x^{n} \mid x \in G\}$ is a normal subgroup of $G$. $G$ could be easily ...
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Showing two numbers are relatively prime in number fields

Solve $x^3-2=y^2$ in integers. The standard way to solve this problem is to consider the arithmetic of the ring of algebraic integers $\mathcal{O}_{\mathbb{Q}(\sqrt{-2})}$ and to show that ...
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Is an algebra the smallest one generated by a certain subset of it?

Let $X$ be a completely regular topological space and let $BC(X)$ denote the space of bounded continuous complex-valued functions on it. Also, let $C(X,[0,1])$ be the set of continuous functions on ...
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Category Theory textbook (learning through guided discovery Dummit and Foote)

sorry for asking the same question in a slightly different angle, I want a book in Category Theory similar to Dummit and Foote's book in Abstract Algebra. I want it to have tons of examples and ...
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A cyclotomic extension of a finite field generate algebraic closure?

Excuse me, I have a question. Let $k$ be a finite field of characteristic $p.$ Let $k^a$ be its algebraic closure. Let $A$ be the elements $x$ of $k^a$ such that $x^m=1$ for some $m$ where we take ...
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1answer
25 views

$k[x_1,\dots,x_n]/\frak{a}$ is an $k$-algebra of finite type?

Let $k$ be a field and $\frak{a}$$\subset k[x_1,\dots,a_n]$ be an ideal. Can someone explain to me why $k[x_1,\dots,x_n]/\frak{a}$ is an $k$-algebra of finite type?
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45 views

Simple extension fields

If I am correct simple extension fields are extensions generated by one element. I have learned that this means that elements of a simple extension can be written as powers of that element as long as ...
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22 views

How to show this is the minimal polynomial

I'm trying to the following problem. But I can't show some irreducibility of the polynomials. Put $L=\mathbb{C}(X,Y,Z)$, $\omega=\frac{-1+\sqrt{-3}}{2}$. Define two automorphism $\sigma, \tau$ of ...
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121 views

Basic Group Theory question

This is not so much a plea of ignorance, but rather me trying to see whether intuitively I actually understand what is going on in group theory. The question asks What group is ...
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43 views

In a principal ideal domain, prove that every non trivial prime ideal is a maximal ideal. What could be wrong in this approach?

In a principal ideal domain, prove that every non trivial prime ideal is a maximal ideal Attempt: Let $R$ be the principal ideal domain. A principal ideal domain $R$ is an integral domain in which ...
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61 views

$\mbox{Im }A\oplus \ker A^t = V$

Let $A:V\to V$ be an endomorphism of a finite dimensional linear space. It's easy to see that $\mbox{Im }A\cap \ker A^t = 0$. Because if $w = Av\in \ker A^t$, then $0 = \langle A^tAv,v\rangle = ...
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3answers
50 views

Non-isomorphic algebraic structures such that each surjects homomorphically onto the other

Off the top of my head, I cannot think of any algebraic structures $X$ and $Y$ such that each surjects homomorphically onto the other, yet $X$ and $Y$ are non-isomorphic. What are some examples of ...
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34 views

Suppose that $I$ is an ideal of $J$ and that $J$ is an ideal of $R$. prove that if $I$ has a unity then $I$ is an ideal of $R$

Suppose that $I$ is an ideal of $J$ and that $J$ is an ideal of $R$. prove that if $I$ has a unity then $I$ is an ideal of $R$ Attempt: Given that $I$ is an ideal of $J$ which means : ...
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Orientability in algebraic setting

I have the following (it can be very silly) question. Suppose I have a commutative algebra $A$ over a field $k$ of $char(k)=0$ which defines a $n$-dimensional smooth variety $X=Spec(A)$. What ...
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Need help with this exercise about real division algebra

I am trying to solve the following exrcise: Prove that complex multiplication does not extend to a multiplication on $\mathbb R^3$ so as to make $\mathbb R^3$ into a real division algebra. I ...
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How many units are there in $ \mathbb{Z}[i\sqrt{2}]/2^{2012} $ [on hold]

How many units are there in $ \mathbb{Z}[i\sqrt{2}]/2^{2012} $
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totally ordered groups [on hold]

please tell me the definition of totally ordered group and tell me does a totally ordered group has the minimum element? Let (H,$\leq$,+) is a totally ordered group. We define: $x\cdot y=(t, t\leq ...
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identifying $\mathbb H$$^n$ with $\mathbb C^{2n}$

Let $X \in M_n(\mathbb H)$ (Hermetian field). It is possible to make $\mathbb H$$^n$ into a $2n$-dimensional vector space over $\mathbb C$, for example, by embedding $\mathbb C$ into $\mathbb H$ as ...
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Show for each $c$, $\{x + c \Bbb Z \in \Bbb Z/c\Bbb Z\mid x^{c−1} \equiv 1 \pmod c\}$ is an abelian group under multiplication of congruence classes

Show for each $c$, the set $$\{x + c \Bbb Z \in \Bbb Z/c\Bbb Z\mid x^{c−1} \equiv 1 \pmod c\}$$ is an abelian group under multiplication of congruence classes.
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Terminology on algebra.

In Probability textbook, algebra usually defined as follows: A collection $\Sigma_0$ of subsets of $S$ is called an algebra on $S$ if $S \in \Sigma_0$ $F\in \Sigma_0 \Rightarrow F^c \in \Sigma_0$ ...
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Submodules of semi-simple modules

Let $R$ be a ring (with unity, not necessarily commutative) and let $P$ be an irreducible $R$-module. Let $$M=\bigoplus_{i=1}^r P$$ be a direct sum of $r$ copies of $P$, for some $r\geq 1$. Then, $M$ ...
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Generalizations of functions

I'm trying to collection examples of mathematical entities that are generalizations of functions. The use of the word "generalization" here doesn't need to be strict, as in every function is an $X$ ...
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Automorphism of certain f.g. free modules

This is a quick question from Frohlich and Taylor's Algebraic Number Theory, II.4, p 94. Let $R$ be a Dedekind domain with quotient field $K$, $\mathfrak p$ is a non-zero prime ideal of $R$ and ...
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Hochschild cohomology of skew polynomial rings

Definition The skew polynomial algebra over $\mathbb{C}$ is defined as $\mathbb{C}\langle x,y\rangle/(xy-yx+x)$ or alternatively as $\mathbb{C}[x,y,\sigma]$, where $\sigma$ is the automorphism on ...
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generators of groups from exact sequence

Suppose I have a middle term exact sequence of finitely generated abelian groups $G \longrightarrow H \longrightarrow K$. How do I get the generators of $H$ if I know the same for other two groups?
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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 domain page of Wikipedia that $$\mathbb{R}[x,y]/(x^2+y^2+1)$$ is ...
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1answer
49 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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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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56 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 $(6543)(21)$ ...
2
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1answer
74 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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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
27 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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71 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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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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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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$(u+1)(u+2)\cdots(u+p)=u^p-u$

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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37 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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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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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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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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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.