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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Annihilator of a flat ideal

Let $R$ be a commutative ring and let $I$ be a finitely generated flat ideal of $R$. Let $J=\mathrm{Ann}(I)$. How can one prove that $I\cap J=0$? This can be found as a remark in the paper of ...
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1answer
18 views

If $R/P$ is an integral domain then $P\vartriangleleft R$ is prime. [duplicate]

Let $R$ be a ring and let $P$ be a proper ideal of $R$. If the quotient ring, $R/P$ is an integral domain then $P\vartriangleleft R$ is prime. For $x,y\in R$ we have $(x+P)(y+P)=xy+P\in ...
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0answers
14 views

CharK=0 (or p) iff CharF=0 (or p), F is subfield of K [duplicate]

Let $F$ be a subfield of the field $K$. Prove that: 1) $CharK=0 \iff CharF=0$ 2) $CharK=p \iff CharF=p,\ p$ is prime. My thoughts: (a) $1_K \in K$, so $ CharK=ord(1_K) \ | \ |K|$ from Lagrange. If ...
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1answer
20 views

Fraction rings ideals members

Let $R$ be a ring with fraction ring $R_S$ and ideal $I$. I saw in arguments that when $a/s$ is in $I_S$ they dont say $a$ is in $I$. Instead they say $a/s=b/t$ with $b \in I$. Why? Many thanks.
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1answer
59 views

Rings where $ab=0$ for all elements

Let $R$ be a ring, not necessarily unital, such that $ab=0$ for all $a,b\in R$. Suppose $R$ only has trivial right ideals. Is it true that $R$ has finite order? Are these rings special?
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1answer
20 views

Order of an orbit of Frobenius action on a algebraically closed field of characteristic p

Consider the action of the Frobenius homomorphism $F^{2}:\,\overline{\mathbb{F}_{q}}\rightarrow\overline{\mathbb{F}_{q}},\,x\rightarrow x^{q^{2}}$ over $\overline{\mathbb{F}_{q}}$ . Let $s=\left\{ ...
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1answer
21 views

Iff $\{s t^{-1}: s, t \in T \} = G$ for a group $G$ and a nonempty subset $T$, then each right coset $Ng$ of $G$ is already trivial

Let $G$ be a group and $T \subseteq G, T ≠ \emptyset$. We now want to consider the set $T T^{-1} := \{s t^{-1}: s, t \in T \}$. I now want to show that $\langle T T^{-1} \rangle ≠ G $ iff there ...
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0answers
8 views

irreducibility test for multinomials over a finite field

I am working in an algebraic cryptosystem, and I need, in the process, ensure that a 3-variables polynomial in a finite field is irreducible but I can't find a practical method to do that. Do you know ...
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1answer
17 views

For any permutation $ \sigma \in S_n$, $(σ(1) − 1)(σ(2) − 2) . . . (σ(n) − n)$ is even when $n$ is odd [on hold]

Let σ be a permutation of ${1, 2, 3, . . . , n}$, n odd. I want to show that $(σ(1) − 1)(σ(2) − 2) . . . (σ(n) − n)$ is even. Thank you.
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1answer
22 views

Finding a finite $p$-group of nilpotency class $n$ for each $n>1$

There's a problem in Rotman's group theory book that goes For $n\ge1$, let $G_n$ be a finite $p$-group of class $n$. Define $H$ to be the group of sequences $(g_1,g_2,\dots)$ for $g_n\in G_n$ and ...
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2answers
33 views

Transforming a Polynomial to Show Irreducibility Using Eisenstein's Criterion

I have a particular polynomial $$z^5-5z^4+30z^3-150z^2+465z-725$$ A quick check in mathematica shows that this polynomial is irreducible over the rationals, however, it does not pass the third ...
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1answer
29 views

Regarding taking powers of prime ideals in a ring

My question is simple to ask: given some prime ideal $P$ in a ring $R$, we can talk about $P^2, P^3$ etc. but can we discuss $P^0$? Is there a convention that says $P^0 = R$, or is there something ...
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2answers
43 views

Find all the homomorphisms from $D_8 \to \mathbb{C}^\times$

Find all of the homomorphisms from $D_8$ to $\mathbb{C}^\times$. So far I have: $\phi : D_8 \rightarrow \mathbb{C}^\times$ $\phi(a)^4 = 1$ so $\phi(a) = \pm 1, \pm i$ $\phi(b)^2$ = 1 so ...
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1answer
16 views

Simplify $\langle \operatorname{Ind}^G_1 1, \operatorname{Ind}^G_H\phi\rangle_G$

Let $G$ be a finite group and $H$ a subgroup. Let $\phi$ be an irreducible character of $H$ and $\mathbb 1$ the trivial character of the trivial subgroup $1$. Let $\langle,\rangle_G$ be the usual ...
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1answer
47 views

$Aut(G)$ is abelian if and only if $G$ is cyclic. [duplicate]

Problem says: Let G be an abelian group. Prove that Aut(G) is abelian if and only if G is cyclic. And I solved $(\Leftarrow)$ direction as follow: Suppose that $G$ is cyclic. Then ...
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0answers
25 views

Pronormality of Sylow subgroups

I need help on proving that every Sylow subgroup of a finite group is pronormal. A subgroup $H$ of a group $G$ is said to be pronormal if for each $g\in G$, the subgroups $H$ and $gHg^{-1}$ are ...
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2answers
66 views

Show that there is odd number of elements of a finite group satisfying $x^3=e$

Show that: Show that there is odd number of elements of a finite group satisfying $x^3=e $? And even number of elements satisfying $x^2\neq e$??? I donot have any idea how to start.
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0answers
22 views

Complex extension isomorphic to $\mathbb{R}$?

Let $K$ be some field extension of $\mathbb{Q}$ containing some complex number $c=a+bi$ with $a,b\in \mathbb{R}$ and $b\neq 0$. Is it possible that $K\cong \mathbb{R}$ as fields? I tried to disprove ...
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0answers
15 views

Describe the prime ideals of Ring $R$ in terms of their generators. [duplicate]

Let $R:=\Bbb C[x,y]$ denote the ring of polynomials in the variables $x$ and $y$, with complex coefficients. Describe the prime ideals of $R$ in terms of their generators. Prime ideals are ideals ...
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1answer
40 views

SHow that $\mathbb{Q}(i,\sqrt{3}) = \mathbb{Q}(i+\sqrt{3})$ [on hold]

I am stuck on how to solve this problem, any hints on what would be useful here?
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2answers
32 views

Is the kernel of the induced homomorphism of a group action of G on a subgroup H a subgroup of H?

If we have a subgroup H of a group G, and act by left regular action on the set A of left cosets of H in G. This induces a homomorphism $\phi: G \rightarrow S_{n}$, where ker($\phi$) is a normal ...
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3answers
56 views

Show that two rings are not isomorphic

I don't know how to show (or why) $M_{2\times2}\mathbb{(R)}$ is not isomorphic to $\mathbb{R}[x]/(x^4-1)$ does it have something to do with the order of coset representatives of the quotient group? ...
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1answer
23 views

Order $4$ subgroup of alternating group $A_4$

I ran into the following problem: Let $H$ be the subgroup $H = \{e, (1\,2)(3\,4), (1\,3)(2\,4), (1\,4)(2\,3)\}$ in $G = A_4 = H \cup \{(1\, 2\, 3), (1\, 3\, 2), (1\, 2\, 4), (1\,4\,2), ...
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1answer
34 views

Show that every polynomial of degree $1,2,$ or $4$ in $\mathbb{Z}_2[x]$ has a root in $\mathbb{Z}_2[x]/(x^4+x+1)$.

The problem: Show that every polynomial of degree $1,2,$ or $4$ in $\mathbb{Z}_2[x]$ has a root in $\mathbb{Z}_2[x]/(x^4+x+1)$. My attempt: I know that the polynomials $x$ and $x+1$ have ...
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2answers
30 views

If $a$ and $b$ are nonzero integers such that each is a divisor of the other, show that $a = ± b$ .

I tried many approaches to this problem. I believe that if I did $b|a=m$ and $a|b=n$ and set $m=n$, then $a$ and $b$ would be equal. Is that how it should be done? If not, please help me out. Thanks.
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What do you call the category of groups with surjections (injections)?

Take the category $Grp$ and throw out all morphisms that are not surjections. I think what's left is still a category. The composition of two surjections is surjective, and identities exist ...
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1answer
20 views

Torsion in module over the ring of convergent power series

I need to understand a passage from a paper which I don't quite understand. Let $M$ be a module over the ring $\mathbb C\{t\}$ of convergent power series. We want to show that $M$ is torsion-free, ...
2
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1answer
65 views

Is $\operatorname{Mat}(2,\mathbb{R})$ isomorphic to $\operatorname{Mat}(3,\mathbb{R})$? [on hold]

As the title. Is $\operatorname{Mat}(2,\mathbb{R})$ isomorphic to $\operatorname{Mat}(3,\mathbb{R})$? I cannot seem to find any bijection that will do. Thanks. $\operatorname{Mat}(n,\mathbb{R})$ ...
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1answer
29 views

Define a new addition ⊕ and multiplication on Z by a⊕b = a + b−1 and ab = a + b−ab.

a+b and ab are the usual integer addition and multiplication. You can assume that this new operation forms a ring, say R is the set of integers with these operations. Then does R have zero-divisors? ...
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0answers
44 views

Isomorphism theorems for topological groups

I know that the second isomorphism theorem for groups doesn't hold for topological groups, the version that I have for the second isomorphism theorem is: If $G$ is a group, $H$ a subgroup of $G$ and ...
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What is the purpose of homomorphisms?

I know that a mapping $\phi:A\to B$ is a homomorphism provided that $$\phi(A*B)=\phi(A)\times\phi(B)$$ where $*$ and $\times$ are two operators on the algebraic structures $A$ and $B$ respectively. In ...
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1answer
30 views

Algebra and field irreducibility [on hold]

I am having trouble with a series of algebra questions on fields. Let $f$ in $F[x]$ be an irreducible polynomial with coefficients in a field $F$ and of degree $\geq 1$. Let $L = F[x]/(f(x))$. ...
2
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1answer
29 views

Algebra, finding the elements of the field and solving irreducible polynomials

I'm trying to do this problem from a practice final but there are no solutions. I honestly am pretty stumped. My thought was since it has 7 elements, then the degree of the polynomial must be one ...
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0answers
35 views

Determine the group of units of a subset of $M_n(\mathbb{C})$

Let $R$ be a commutative ring. Let $R=\bigg\{\begin{bmatrix}u & v\\ 0 & u\end{bmatrix}:u,v\in\mathbb{C}\bigg\}$. Determine the group of units $R^{\times}$ of $R$. My try: Let ...
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0answers
28 views

Bezout relation with integral coefficients

Suppose I have two monic polynomials $f$ and $g$ with coefficients in $\mathbb{Z}$. I also suppose that $f$ and $g$ are coprime as polynomials over $\mathbb{Q}$. In particular, there exists a Bezout ...
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1answer
54 views

Why is $R((X))$ defined as follows?

Let $R$ be a commutative ring. Then $R((X))$ is defined as the set of all $\sum_{n\geq N} a_n X_n$ where $N\in\mathbb{Z}$ and is called "The Formal Laurent series". But why? Why don't we consider ...
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Let G be a group, and suppose that H is a subgroup of G of order 5, and no other subgroup of G has order 5. Show that H is a normal subgroup of G. [on hold]

Let G be a group, and suppose that H is a subgroup of G of order 5, and no other subgroup of G has order 5. Show that H is a normal subgroup of G. Picture of my proof I got 5/10 on this problem. It ...
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need help with abstract algebra [on hold]

Let $A_4 \le S_4$, and $$A_4 = \{ (1), (1 2)(3 4), (1 3)(2 4), (1 4)(2 3), (1 2 3), (1 3 2), (1 2 4), (1 4 2), (1 3 4), (1 4 3), (2 3 4), (2 4 3) \}$$ Find the conjugacy classes and the class ...
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Using Gap system. [on hold]

I'm new at the GAP. Probably I can't use this system, what I type doesn't work. For instance why the following doesn't work? for i in [1..1160] do Print("Processing semigroup number ",i,"\n"); ...
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Does Nagata theorem hold in a field that is not algebraically closed?

Let $R$ be a finitely generated $k$ - algebra and $G$ be a reductive group acting rationally on it. Then a theorem of Nagata says that the invariant ring $R^G$ is also finitely generated. Here $k$ ...
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for every left ideal $L$ of $R$ and $R$-module homomorphism $g:L\to A$, there exists $a\in A$ such that $g(r)=ra$ for every $r\in R$.

Let be a $R$ ring with a identity. An $R$-module $A$ is injective iff for every left ideal $L$ of $R$ and $R$-module homomorphism $g:L\to A$, there exists $a\in A$ such that $g(r)=ra$ for every ...
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1answer
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Center of finite dimensional division $\mathbb{R}$-algebra?

Let $D$ be a finite dimensional division $\mathbb{R}$-algebra. Why is it that $Z(D)=\mathbb{R}$ or $Z(D)=\mathbb{C}$? I have seen an explanation: It is because $\mathbb{C}$ is the only non ...
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0answers
9 views

Endomorphism of Central Simple Algebra

Let $A\in\mathscr{C}(F)$, i.e. $A$ is a central simple algebra. Show that $\text{End}_F(A)\cong M_n(F)$ as $F$-algebras where $n=\dim A$. My idea is to consider $A$ as a $F$-vector space, then ...
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1answer
53 views

About the $1$ of ring

I could not find neither a proof nor a counterexample, can anyone solve this? Let $A$ be a finite dimensional $k$-algebra. (It not necessarily has $1$.) If $$\mu:A\otimes A \rightarrow A,\ ...
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2answers
47 views

Subfield of a finite field

I have started studying field theory and i have a question.somewhere i saw that a finite field with $p^m $ elements has a subfield of order $p^m $ where $m$ is a divisor of $n $.My question that if ...
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1answer
21 views

Isomorphism between the group $(\mathbb Z[x], +)$ and $(\mathbb Q_{>0}, .)$ [duplicate]

In one of my assignment, I was told that $(\mathbb Z[x], +)$ and $(\mathbb Q_{>0}, .)$ are isomorphic, with $$\phi (\sum_{k=0}^n a_k x^k) = \prod_{k=0}^n p_k^{a_k}$$ is a one-to-one surjective ...
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1answer
50 views

Subgroups of $\mathbb{Q}/ \mathbb{Z}$ [on hold]

I was asked to show in my exam that all the subgroups of $\mathbb{Q}/ \mathbb{Z}$ are of the form $\{\mathbb Z + a/p^i\}$ where $p$ is a prime number and $0\leq a \lt p^i$, where $i$ varies over all ...
0
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1answer
18 views

Confused about order in Opposite Algebra

I am facing confusion in the "order of multiplication" regarding the opposite algebra $B^o$ in the following working: Define a right $B^o$-module $M_\phi$ where $M_\phi=M$ via $m\cdot b=m\phi(b)$. ...
0
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1answer
15 views

Problem on field extension related to irreducible polynomial

Suppose $\gamma,\gamma'\in\Bbb C$ are distinct roots of the same irreducible polynomial $p\in\Bbb Q[x]$. Suppose $x^2-5$ is irreducible in $\Bbb Q(\gamma)[x] $. Show that it is also irreducible in ...
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0answers
33 views

if $\phi : G \mapsto G'$ $y \mapsto \left[ x,y \right]$ why $\phi$ is onto? [on hold]

if $\phi : G \mapsto G'$ $y \mapsto \left[ x,y \right]$ why $\phi$ is onto?