Questions tagged [abstract-algebra]

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16 views

Confusion about direct sums of $\mathbb{Z}/pq\mathbb{Z}$ where $p, q$ prime

I think I have some kind of fundamental misunderstanding of direct products and direct sums of groups. I'm trying to understand the following statement: For prime numbers $p \neq q$, we have $\mathbb{...
-2
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0answers
21 views

Prime Ideal of R[x] then intersection P with R is a prime ideal of R [closed]

If $P$ is a prime ideal of $R[x]$. Show that $P∩R$ is a prime ideal of $R$. I understand the case when R is commutative i.e. $R[x]$ is commutative. But when $R$ is a non commutative ring then what ...
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2answers
31 views

Group of units in a finite field

Let $f(x) \in \mathbb{Z}/3\mathbb{Z}[x]$ be a cubic irreducible polynomial and let $F = \mathbb{Z}/3\mathbb{Z}[x]/(f(x))$. I want to show that either $x$ or $2x$ generates the group of units of $F$. ...
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0answers
18 views

Find the subset of the 4x4 matrix [closed]

The matrix looks like 0 0 0 1 0 0 1 0 1 0 0 0 0 1 0 0 I have to find the subsets but I'm a little confused on where to start.
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1answer
16 views

Taking sheaf of union of algebraic open subsets is same as intersection of sections

Let $I(V) = \{ f \in k[x_1, \dots, x_n] : f(P) = 0, \forall p \in V\}$ be the vanishing ideal of $V$, and let $k[V] = k[x_1, \dots, x_n]/I(V)$ be the coordinate ring of $V$ (ring of regular functions)....
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3answers
34 views

The equation $2x=a$ has no solution in an infinite additive cyclic group generated by $a$

Q: If $G$ is an infinite additive cyclic group with the generator element $a$, prove that the equation $2x=a$ has no solution in $G$, $x \in G $. Answer: Since $G$ is an infinite cyclic group then $|...
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11 views

What is an example of a Hilbert module that is not a Hilbert space?

Hilbert modules are always considered over some fixed C*-algebra of coefficients in which the generalised inner product takes values, so we cannot really say what a Hilbert module is without ...
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1answer
27 views

Proof that $\sqrt{\det(XY)^2} \leq \sqrt{\det(Y^TY)\det(XX^T)}$?

Proof that $\sqrt{\det(XY)^2} \leq \sqrt{\det(Y^TY)\det(XX^T)}$? I feel like I could use Cauchy-Binet here but I'm unsure of how to do so. Any thoughts?
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2answers
28 views

Show that $\text {Hom}_{\mathbb R[x]}(M,N)=\{0\}$ where $M,N=\mathbb R^2$ are $\mathbb R[x]$-modules where $X$ acts as $A$ in $M$ and $B$ in $N$

Let $A=\begin{pmatrix}2&0\\0&3\end{pmatrix}$ and $B=\begin{pmatrix}1&0\\0&0\end{pmatrix}$ Show that $\text {Hom}_{\mathbb R[x]}(M,N)=\{0\}$ where $M,N=\mathbb R^2$ are $\mathbb R[X]$-...
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Gramian Matrix of General Coxeter Group and the sign of its determinant

Suppose $(W,S)$ is a Coxeter Group with defining relations $(ss')^{m_{ss'}}=id$ for all $s,s'\in S$. Let $K$ be the matrix indexed by $S\times S$ such that $K_{ss'} = -2\cos(\frac{\pi}{m_{ss'}})$ (...
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0answers
23 views

Is it always possible to factor an element that is not irreducible into two non-units?

A book I am reading defines irreducible elements of an integral domain $D$ as follows: A nonzero element $a\in D$ is called an irreducible if $a$ is not a unit and, whenever $b$, $c \in D$ with $a=...
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1answer
19 views

Dimension of irreducible components of $f^{-1}(X)$, where $X$ is a hypersurface

I am trying to understand the following problem from Gathmann's algebraic geometry notes:If $f:\mathbb{P}^n\rightarrow\mathbb{P}^m$ is a morphism and $X\subset\mathbb{P}^m$ is a hypersurface then ...
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0answers
18 views

Are there integral domains that don't satisfy ACCPI but where there is (unique) factorization? [duplicate]

If an integral domain $R$ doesn't satisfy the ascending chain condition for principal ideals, then factorization of an element of $R$ (into irreducible elements) might fail: if you choose the wrong ...
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0answers
19 views

Show that all composition series of Z60 are isomorphic [closed]

Composition series are subnormal series and subnormal series are isomorphic if there is a one one correspondence between the collection of factor groups such that corresponding factor groups are ...
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0answers
24 views

For which $r\in\mathbb{Q}$ is $\mathbb{Q}[x]/\langle x^2 - r\rangle$ a field? [duplicate]

So I know that $\mathbb{Q}[x]$ is a P.I.D. Does this mean that I only have to find where $x^2-r$ is irreducible over $\mathbb{Q}[x]$? I have tried approaching this by doing the following: $x^2-r$ can ...
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0answers
20 views

Shortcurt in finding conjugancy classes?

I have to find all the conjugancy classes of $A_4$ when $y=(234)$ is a cycle and the transposition $x=(12)(34)$ in $S_4$ which satisfies $y^3=x^2=(xy)^3=I$, is there any clever tricks/shortcurts or ...
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1answer
19 views

Asumptions required for Inverse Limit on Infinite Galois Group

I am currently trying to get some knowledge on Infinite Galois Theory. During the construction of the Galois Group of an infinite Galois extension through an inverse limit of Galois group of ...
3
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1answer
66 views

Prove that $\mathbb{Q}(\sqrt{2},\sqrt[3]{3})=\mathbb{Q}(\sqrt{2}\sqrt[3]{3})=\mathbb{Q}(\sqrt{2}+\sqrt[3]{3})$

Showing that $\mathbb{Q}(\sqrt{2},\sqrt[3]{3})$ has degree 6 over $\mathbb{Q}$ is straighforward: It contains $\mathbb{Q}(\sqrt{2})$ and $\mathbb{Q}(\sqrt[3]{3})$ which are degree 2 and 3 over $\...
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0answers
13 views

Complex and real vectors [closed]

Using Euler's rule ,the division of two vectors in complex space is allowed , but why it is not in real space?! Plus , if we are to represent both spaces , are they going to be related ?
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1answer
51 views

Morphisms from quotient ring

Let $f(x)$ be a polynomial in $\mathbb{Z}[x], \left<f(x)\right>$ be the ideal it generates. Let $R$ be a ring. Prove that giving a ring homomorphsim $\mathbb{Z}[x]/\left<f(x)\right> \...
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1answer
38 views

The fundamental aspects of the square root [closed]

When I was in High School learning algebra we came upon solving for roots. When doing this for a quadratic you sometimes end up having square roots in your answer. Due to uncertainty we cannot ...
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0answers
24 views

Iterated tensor product of a ring

Considering a ring $R$ as a $\mathbb{Z}$-algebra, is there anything known about the sequence $$ R, \:\: R \otimes_{\mathbb{Z}} R, \:\: R \otimes_{\mathbb{Z}} R \otimes_{\mathbb{Z}} R, \:\: ... $$ ? I ...
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1answer
21 views

Example of torsion modules over integral domain that has zero annihilator

I was able to prove that if $R$ is an integral domain and $M$ is a finitely generated $R$-module is torsion if and only if $Ann(M) \neq 0$, but so far, I have failed to come up with an example that ...
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0answers
41 views

Connecting the various interpretations of normal subgroup

What is a normal subgroup? I have heard all of the following: A subgroup $N$ of $G$ is normal if $gNg^{-1} = N$ for all $g \in G$. Or equivalently, a subgroup is normal if it is invariant under ...
3
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1answer
60 views

grothendieck group over flat family

Say $f:X\to Y$ is a flat family of finite length objects. If $K_0(f^{-1}(y))$ stays constant for all $y$ in an open dense set of $Y$, I believe it is then true that $K_0(f^{-1}(y))$ stays constant for ...
2
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0answers
20 views

Basic question about inseparable field extension

Suppose $L/K$ is not separable. Does this imply that there is an $a\in L$ such that $a^p\in K$ but $a\not\in K$? I know that the characteristic of the fields must necessarily be $p>0$. By ...
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1answer
47 views

Möbius transformation preserving polynomial

Let $K$ be an algebraically closed field, and $x$ transcendent over $K$. We know that the group of $K$-automorphism of $K(x)$ is $\operatorname{PGL}(2,K)$, whose elements satisfy $ \alpha (x)= \...
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0answers
16 views

Giving explicit proof for the tensor product [closed]

please help solving next question. Give the explicit proof that $(a,b,c) \rightarrow (a \otimes b) \otimes c$ is a universal trilinear function. Thanks in advance.
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0answers
46 views

Action of the symmetry group of a cube on opposite faces defines a surjective homomorphism from $S_{4}$ to $S_{3}$

Show that the action of the symmetry group of a cube on pairs of opposite faces defines a surjective homomorphism from $S_{4}$ to $S_{3}$. I am stuck on this problem and don't know where to begin. I ...
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1answer
27 views

How to show surjectivity in this task?

Let $R$ be a commutative ring with unity and $e$ an idempotent, $e\neq 1, e\neq0$. I need to show that $Re$ and $R(1-e)$ are subrings with unity and $R \cong Re \times R(1-e)$. Notice that $(1-e)$ is ...
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3answers
21 views

Order of elements of the group of $3 \times 3$ upper-triangular matrices over $\mathbb{Z}/p\mathbb{Z}$ with $1$'s in the diagonal

Suppose $p$ is an odd prime. Let $G$ be the group of $3 \times 3$ upper-triangular matrices over $\mathbb{Z}/p\mathbb{Z}$ with $1$'s down the diagonal. Show that every element of $G$ has order that ...
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1answer
40 views

Why does $2$ ramify in $\mathbb{Z}[i]$ but not 3?

I know that we can write $(2)=(1+i)^2$, and from this question we have that $\mathbb{Z}[i]/(3)$ is a field. But perhaps I'm confused why this is so. In particular, doesn't the same computation in the ...
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2answers
33 views

Can I use the fact that {0} is a subring of every ring in this statements?

Prove: a) A subring $S$ of a ring with unity $R$ does not have to be a ring with unity. b) A subring $S$ of a ring $R$ without unity can contain a unity. c) A subring $S$ of a ring $R$ can have a ...
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1answer
39 views

Let $\sigma, \tau \in S_n$. Prove $\sigma\tau \text{ is even} \iff \sigma \text{ and }\tau$ are both even or both odd.

I know that this has to be proven both ways. If $\sigma \tau$ is even then $\sigma$ and $\tau$ are both even or both odd, and if $\sigma$ and $\tau$ are both even or both odd then $\sigma \tau$ is ...
2
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1answer
39 views

Suppose that $G$ is a group and $a\in G$ is given with $|a|=100.$ What is $|a^{65}|$?

Suppose that $G$ is a group and $a\in G$ is given with $|a|=100.$ What is (the numerical value of) $|a^{65}|$? Would it be $100/5$ (where $5 = \gcd(100,65)$)?
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0answers
33 views

$|GL(n,\mathbb{Z}_{p^{k}})|$

I want to calculate the order of the group $G = GL(n,\mathbb{Z}_{p^{k}})$ consist of all invertible matrices size $n\times n$ with coefficients in $\mathbb{Z}_{p^{k}}$ with $p$ a prime number, $k\in\...
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0answers
20 views

Finding an orthogonal basis using Gram matrix

In $R^3$ scalar product is set using Gram matrix: $$ G(i, j, k) = \begin{pmatrix} 2 & 1 & 0 \\ 1 & 2 & 1 \\ 0 & 1 & 3 \\ \end{pmatrix} $$ Find an ...
1
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2answers
55 views

Given $x^{p^2} = 1, x^p = y^p, yxy^{-1}=x^{p+1}$, show that $(yx^{-1})^p=1$

Suppose $p$ is an odd prime. Given $x^{p^2} = 1, x^p = y^p, yxy^{-1}=x^{p+1}$, show that $(yx^{-1})^p=1$, where $1$ is the identity. I know that the conjugate of power is the power of conjugate, so $...
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1answer
24 views

Let $G$ be solvable. Let $g\in G$. What can we say about $g=[g^{x_1}, g^{y_1}]\dots [g^{x_n}, g^{y_n}]$ with $x_i,y_i\in G,i=1,…,n$?

We have $G$ a solvable group. Let g be an element of G. What can we say about $$g = [g^{x_{1}}, g^{y_{1}}]\cdot\cdot\cdot[g^{x_{n}}, g^{y_{n}}]$$ with $x_{i}, y_{i} \in G, i = 1,..., n$?
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2answers
29 views

$Ind_H^G(\pi\oplus \mu) \simeq Ind_H^G(\pi)\oplus Ind_H^G(\mu)$

Let $\pi$ and $\mu$ be representations of a subgorup $H\leq G$. ($G$ a finite group.) My question is whether or not it is true that $$ Ind_H^G(\pi\oplus \mu) \simeq Ind_H^G(\pi)\oplus Ind_H^G(\mu) $$ ...
0
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1answer
30 views

Are these a particular case of the universal property of the coproduct and product?

I was talking to a friend about the results presented here and here. He told me that these are particular cases of the universal property of the coproduct and of the product (in a category). I have ...
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0answers
39 views

Definition of an algebra over a field

My professor defined an algebra over a field as a vector space over that field that is also a ring. He did not state however with regard to what operations is the algebra a ring so can I assume the ...
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1answer
33 views

Direct sum, product, sum and intersection of ideals

Is is true that in general the direct sum of an ideal of a ring with itself is equal to the square of that ideal (i.e. the product with itself). More generally, is $I \oplus J = IJ$ for I,J ideals? ...
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2answers
28 views

Is it true that a non-principal prime ideal in k[x,y] must have a pair of coprime elements? [closed]

As the title suggests. It is a step towards characterising prime ideals in k[x,y]
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0answers
18 views

Prove that a specific $SL(3,\mathbb{Z})$ element has order three using defining relations

Consider the following element in $SL(3,\mathbb{Z})$: $$X=T_{31}^{-k}T_{13}^2T_{31}^{-m}T_{13}^{-m-1}T_{31}^{2}T_{13}^{-n}T_{31}^{-4m^2-3}S_{13}^{-1}T_{23}^{2m^2+m+1-(4m^2+3)n}T_{21}^{k-n}S_{23}.$$ ...
2
votes
1answer
44 views

Algebra projective over subalgebra

Let $A$ be an algebra, $S \subset A$ a subalgebra.* I was wondering: Is $A$ projective as an $S$-module? I have a feeling that this cannot be true in this generality (the feeling comes from the ...
1
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1answer
27 views

General question about finding the minimal polynomial

Suppose you're given a root $\alpha$ and asked to find the minimal polynomial of $\alpha$ over $F$, where $F$ is some field. The usual strategy I've seen for solving these problems is to set $x=\alpha$...
-2
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1answer
50 views

Show that $GL(2, \Bbb Q)$ is isomorphic to a subgroup of $GL(3, \Bbb Q)$ [closed]

What should be my approach for this particular question and general approach to prove isomorphism of one group to another ?
8
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2answers
97 views

If a forgetful functor between varieties preserves coproducts, does it have a right adjoint?

Suppose that $C,D$ are two varieties (equational classes) and that $U:C\rightarrow D$ is the forgetful functor. Is it true that if this fuctor preserves coproducts and initial objects then it has a ...
1
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1answer
32 views

How to prove that $G$ is semidirect product?

If $ N\triangleleft G $ and $ G / N $ is a free group, I want to show that $ G $ is a semi-direct product of $ N $ by $ G / N $, but I don't see how to prove it, I know if $ N $ is normal to $ G $, ...

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