Questions tagged [abstract-algebra]

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17 views

How to show this?(existence of the element)[Galois theory]

There are fields $\mathbb{Q}$, $L$ and $M$ $s.t.$ $\mathbb{Q} \subset L \subset M$ Here the $\alpha \in L$ and $[L ; \mathbb{Q}] = 2$ The field $M(= L(\sqrt \alpha) )$ is a finite normal extension ...
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2answers
21 views

Expressing a 3-cycle in $S_5$ as a product of two $5$-cycles.

In $S_5$ prove that any $3$-cycle is a product of two $5$-cycles. My arguments are as follows: There are $24$ five cycles in $S_5$ and they generate $A_5$ since $A_5$ is Simple. All $3$-cycles also ...
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0answers
8 views

Terminology Question: “Local Module” : Modules with a unique maximal submodule.

We call a ring with a unique maximal (left/right) ideal $\textit{local}$. Do we have common terminology for modules that have a unique maximal submodule? It would seem to make sense to call them local ...
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15 views

Reduced quadratic forms, reduced ideals, bounds of coefficients

I have a question about the connection between ideals an forms. Let $f =(a,b,c)$ be integral primitive positive definite binary quadratic forms and $\Delta$ be the discriminant. It is clear, that if $...
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2answers
21 views

Let the set $S=\{f \in C[a,b] : f(x)=0 \forall x \in [c,d]\}$ in $C[a,b]$

Let $a,b,c,d$ be a real number such that $a<c<d<b$ .consider the ring $C[a,b]$ with pointwise addition and multiplication Let the set $$S=\{f \in C[a,b] : f(x)=0 \forall x \in [c,d]\}$$ ...
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1answer
18 views

The finite subgroups of the multiplicative group of some division ring act semiregulary one some finite ablian group

In these lecture notes about classical groups, on page 4, 2nd paragraph, the finite subgroups of the multiplicative group of some division ring/skew field are considered: Let $G$ be a finite ...
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1answer
20 views

Show that the ideal generated by a polynomial and a prime is maximal [duplicate]

I need to show that the ideal generated by $x^2 - x + 1$ and 17 in $\mathbb{Z}[x]$ is maximal As far as I know, it would be sufficient to show that the polynomial is irreducible mod 17, what could be ...
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0answers
20 views

automorphism groups of the orthogonal group and special orthogonal group

What are the automorphism groups of the groups $SO(3)$ and $O(3)$, that is, the 3x3 orthogonal matrices and the 3x3 orthogonal matrices with determinant 1. Thanks
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2answers
54 views

Irreducibility of Polynomials without Eisenstein

In order to do a different task, I need to show that $f:=X^4-10X^2+1$ is irreducible over $\mathbb{Q}$ and that $g:=X^2Y^2-2Y^2-3X+6$ is irreducible over $\mathbb{Q}[X]$. Since I don't think that ...
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2answers
29 views

$A[x]$ is a commutative domain if and only if $A$ is a commutative domain [duplicate]

Starting with assuming $A[x]$ is a domain I only take the elements of $A$ and view them as polynomials of degree zero and use the hypothesis My problem is going the other way: Let $A[x]$ be a domain,...
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2answers
46 views

Is it true that $\mathbb Z[x_1, \ldots, x_n] / \mathcal P \simeq (\mathbb Z/ p) [ x_1, \ldots, x_n]$?

Here $\mathcal P$ is a prime ideal of $\mathbb Z[x_1, \ldots, x_n]$ and $p = \mathcal P \cap \mathbb Z$ (and assume $p \neq (0)$). I think I can prove this using the 3rd part of the second isomorphism ...
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1answer
33 views

Equating the coefficients of the following equation

How to equate the coefficients of $$\sum_{k=0}^{\infty}\sum_{n=0}^{\infty}P(n,k)\frac{t^n}{n!}\frac{x^k}{k!}=\sum_{m=1}^{\infty}\left(\sum_{k=0}^{\infty}\sum_{n=1}^{\infty}Q(n)\frac{t^n}{n!}\frac{x^...
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1answer
42 views

Show that $y-x^2 \notin \langle x(x-1), (x-1)y\rangle$ in $\mathbb{C}[x,y]$

I am in the middle of an exercise and I am stuck in a final step in which I want to show that $y-x^2 \notin \langle x(x-1), (x-1)y\rangle$ in $\mathbb{C}[x,y]$. My first thought was that $x-1$ does ...
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1answer
27 views

In Maschke's theorem, does the averaged projector $\bar P_U$ equal $P_U$?

Consider a finite group $G$ represented on a $\mathbb K$-vector space $V$ with $\operatorname{char}\mathbb K$ not dividing $|G|$, and let $U$ be an invariant subspace for the representation, and $W$ ...
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1answer
50 views

Modifying the definition of a prime ideal in a commutative ring.

According to JOSEPH A.GALLAIN EDITION 8th A prime ideal $A$ of a commutative ring $R$ is a proper ideal of $R$ such that $a, b \in R$ and $ab \in A$ imply $a \in A$ or $b \in A$. I am confused ...
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1answer
27 views

Dual of the map $f: A \rightarrow eAe$.

Let $A$ be a finite dimensional algebra over a field $k$ and $e$ be an idempotent in $A$. Consider the $k$-linear map $f : A \rightarrow eAe$ such that $f(a)=eae$ for $a \in A$. If we take the $k$-...
2
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2answers
48 views

Conceptual question about the strategy used in the following theorem: Every Ideal of $F[x]$ is Principal

To refresh everyone, the following picture from Pinter's "A Book of Abstract Algebra" details the proof for the theorem that Every Ideal of $F[x]$ is Principal: This strategy is pretty common, so I ...
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1answer
29 views

Proving $|\operatorname{Aut}(E/F)|\leq\left[E:F\right]$ for $E$ splitting field [duplicate]

Let $F$ be a field and $E$ be a spliting field of some $f(x)\in F\left[x\right]$, how does one prove that $$|\operatorname{Aut}(E/F)|\leq\left[E:F\right]?$$ I have no idea since I know $\...
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0answers
22 views

Proving that liouville's constant is transcendental with guided steps

So i got this guided question that basically proves why liouville's constant is transcendental. Here is the Question: let $\alpha=\sum_{k=1}^\infty 10^{-k!}$ be liouville's constant, in this drill we ...
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1answer
32 views

Injective homomorphism between two finite groups with the same order

I want to show given two groups $G,H$ and the homomorphism $f:G\rightarrow H$ injective with $\text{ord}(G)=\text{ord}(H)<\infty$, then $f$ is bijective. Thank you very much in advance.
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1answer
39 views

The order of the Center of the given Group

Given that $$\displaystyle G=\{e,x,x^2,x^3,y,xy,x^2 y,x^3y \}$$ with ${\rm ord}(x)=4$ and ${\rm ord}(y)=2$ such that $xy=yx^3$ Then find the number of elements in the center of the group $G$ ...
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3answers
33 views

Uniqueness of a normal subgroup

If I know that a subgroup of a finite group is a normal subgroup, is this condition sufficient to state that it is unique? I think that is sufficient thanks to Sylow's theorems, but in any exercises I'...
3
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1answer
44 views

$\gcd(135-14i, 155+34i)$ with the Euclidean algorithm

I'm trying to find $\gcd(135-14i, 155+34i)$ in $\mathbb{ℤ}[i]$ with the Euclidean algorithm, but at some point I get stuck. I think it is because I don't have a systematic way of finding suitable ...
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1answer
35 views

About a map between $A$ and $eAe$.

Let $A$ be a fin. dim. $k$-algebra and $e \in A$ be an idempotent. Let $f: A \rightarrow eAe$ be a map such that $f(e_i)=e_i$ if $e_i \in eAe$ $f(e_i)=0$ otherwise. I think it may not be an ...
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1answer
53 views

Group and subgroup proof [duplicate]

I've been trying to teach myself some group theory and have come a across that has really stumped me. It is as follows: Prove that in any finite group $G$ with a subgroup $H$ of order exactly half ...
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1answer
29 views

Reference request: Axiomatic treatment of multiplicative functions?

I'm currently reading Apostol's analytic number theory, Chapter 2 on multiplicative functions. While the current exposition is nice, I can't help but feel that there has to been some algebraic ...
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1answer
57 views

Representation over a finite field

Are there any two inequivalent and irreducible $F$-representations of a finite group $G$ (where $F$ is a field of positive characteristic) having the same characters? I can surely find an example in ...
2
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2answers
68 views

If a group $G$ has order $1001$, prove that all subgroups are normal.

If $\left|G\right| = 1001= 7\cdot11\cdot13$, we want to prove that all possible subgroups exists, and all of them are normal. My first idea is to apply Sylows' Theorems. For subgroups with order $7,...
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1answer
21 views

Action of a Subgroup of Symmetric Group

Let H be a Subgroup of $S_n$ having the action on {1,2,...n} transitive. Moreover if H is generated by Transpositions, I need to show that H is the whole group. What I am thinking is that If H is not ...
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2answers
55 views

How to show $x \otimes 1 = 0 \in R \otimes_{k_0} k$ implies $x=0$?

Let $k_0$ be a field, $k$ a field extension of $k_0$ and $R$ a $k_0$ algebra. I would like to deduce that $x \otimes 1 = 0 \in R \otimes_{k_0} k$ implies $x=0$. I asked a similar question in the past ...
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4answers
61 views

Idempotent, commutative and invertible

Is there a mathematical class and structure in which there exist many objects that are distinct, invertible, commutative and idempotent? Like a set of toggle switches with no hysteresis, so the state ...
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0answers
25 views

Looking for mistake in proof that Noetherian ring is Artinian.

Consider a Noetherian ring $R$ and an ideal $I$, whose associated primes are maximal ideals. I want to show that $R$ is Artinian (to conclude that $\frac{R}{I}$ is Artinian). My attempt: I assumed ...
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1answer
29 views

Ring structure on tensor product of rings

I previously asked a question on whether isomorphisms of modules could be somehow translated into isomorphisms of rings, in order to find that $$U^{1}R\otimes V^{-1}R \simeq U^{-1}(V^{-1}R)$$ as ...
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0answers
20 views

Classification of isometry (and similarity transformation) in higher dimensions

As we know, bijective isometry between real normed spaces is affine (Mazur - Ulam 1932). Does there exist some closed forms or classifications of isometries in higher dimensional spaces like what we ...
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1answer
26 views

Hartshorne Lemma II 8.9.

I am trying to understand the proof of Hartshorne's Lemma II 8.9: Let $A$ be a noetherian local integral domain, with residue field $k$ and quotient field $K$. If $M$ is a finitely generated A-...
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0answers
29 views

Is this matrix in Row Echelon Form (REF not RREF)?

$\begin{bmatrix} 1&2&5&8\\0&0&1&3\\0&0&0&1\end{bmatrix}$ Do the diagonal elements of an RREF matrix equal to !? Also, does the matrix below in REF be made? $\...
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1answer
11 views

Showing that a left $R$-module is not completely reducible

Let $M:=\begin{pmatrix} \Bbb C \\ \Bbb C \end{pmatrix}$ be a left $R:=\begin{pmatrix} \Bbb C & 0 \\ \Bbb C & \Bbb C \end{pmatrix}$-module. We would like to $M$ is not completely reducible as ...
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1answer
23 views

How to determine if a matrix can be decomposed into only shear matrices

My question is that given an arbitrary $2\times2$ matrix with unit determinant, say $$ G_r =\left(\begin{array}{cc}\cos(g d) & -\frac{1}{n_0 g} \sin(g d) \\ n_0 g \sin(g d) & \cos(g d)\end{...
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1answer
24 views

Question about basic property of ring homomorphism

I read the book Contemporary Abstract Algebra by Joseph Gallian. In Page 266, theorem 15.1, the third property claims If A is ideal and Φ is onto S, then Φ(Α) is ideal. Where Φ is a ring ...
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2answers
134 views

Solve the Diophantine Equation $x^2 + 7 = y^5$.

This is a duplicate question of Find integers solutions of $x^2+7=y^5$, however there was no full answer. The solutions $(\pm5, 2)$ and $(\pm 181, 8)$ have been found. The usual strategy for such a ...
2
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0answers
48 views

Compute some intermediate extension on $\mathbb{Q}_3( \sqrt[4]{-3}, \sqrt[3]{2}, \xi_4)/\mathbb{Q}_3$

Let $K = \mathbb{Q}_3$ and $L = \mathbb{Q}_3( \sqrt[4]{-3}, \sqrt[3]{2}, \xi_4)$ where $\xi_4$ denotes a $4$-th root of unity. One can show that $\operatorname{Gal}(L/K) \simeq D_4 \times C_3$. ...
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1answer
24 views

Proving Transitivity for the equivalence relation: $\langle a,b \rangle \sim \langle c,d \rangle \iff a+_{\mathbb{N}}d=b+_{\mathbb{N}}c$

For any $a,b,c,d \in \mathbb N$, I am trying to demonstrate that $\langle a,b \rangle \sim \langle c,d \rangle \iff a+_{\mathbb{N}}d=b+_{\mathbb{N}}c$. I am trying to do this without invoking the ...
1
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1answer
42 views

Three-dimensional Heisenberg algebra

In my lecture notes, it's given that The three-dimensional Heisenberg algebra is a Lie algebra whose three basis elements $p,q,c$ satisfy $[p,q]=c$ and $[c,p]=[c,q]=0$ How would one formally check ...
2
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1answer
36 views

Left-invariant and almost right-invariant metric on a Lie group

Suppose I have a (finite-dimensional) Lie group $(G,\circ)$ with identity element $e\in G$. Then I can always construct a left-invariant metric $$ g_q\colon T_qG\times T_gG \to \mathbb [0,\infty),\...
2
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1answer
29 views

Irreducible Polynomials and $F^*$ notation

So in my lecture notes for an irreducible polynomial I have the following definition A non-zero polynomial $f \in F[X]$ is irreducible iff $f \notin F^*$ and if $f=gh$ then either $g \in F^*$ or $h ...
1
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3answers
92 views

How to quotient $\mathbb Z [\sqrt{-11}] / (1+\sqrt{-11})$?

That's it. I want to do this quotient, $\mathbb Z [\sqrt{-11}] / (1+\sqrt{-11})$. My first idea was to see which elements are in the ideal: $(a+b\sqrt{-11})(1+\sqrt{-11}) = a+ a\sqrt{-11} + b\sqrt{-...
2
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3answers
48 views

Find a polynomial with rational coefficients

Let $γ$ be a root of $x^5 − x + 1 = 0$ in an algebraic closure of $\mathbb{Q}$. Find a polynomial with rational coefficients of which $γ +\sqrt2$ is a root Is it possible to directly modify the ...
0
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0answers
41 views

On the primary decomposition of an ideal

Let $I$ be the ideal generated by $x^2-y^3$ and $y^2-x^3$ in $\mathbb{C}[x,y]$. I am trying to answer two questions: What is the composition length of the module $\frac{\mathbb{C}[x,y]}I$. What is ...
0
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2answers
38 views

Does an ideal $I=(x_1,…,x_n)$ which contains a regular element implie $x_i$ is regular for some $i$?

Suppose $R$ is a commutative ring. $I=(x_1,...,x_n)$ is an ideal. Suppose $I$ contains a $R$-regular element. Does this imply that $x_i$ is $R$-regular for some $i$? If not, is this true for ...
3
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0answers
78 views

Alternative proof of the generalized associative law for groups

The generalized associative law for groups claims that the value of $a_1\star a_2\star ... \star a_n$ is independent of how it is bracketed, where the symbols denote the usual notations of group ...