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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Sylow $p$-groups in $GL_3(\Bbb F_p)$

Fix a prime $p$. How many Sylow $p$-groups are there in $GL_3(\Bbb F_p)$? Let $s_p$ be the number of Sylow $p$-groups in $GL_3(\Bbb F_p)$. By the Sylow theorems, $s_p\equiv 1\mod p$ and if $p^e$ ...
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Can the order of $x \in U_{31}$ be $10$?

Does there exist an element $x\in U_{31}$ such that the order of $x$ is $10$? Here $U_{31}$ is the group of units of $\mathbb{Z}/31\mathbb{Z}$.
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Unique order $d$ subgroup of $\Bbb Z/n\Bbb Z$ if $d\mid n$ [duplicate]

Let $G=\Bbb Z/n \Bbb Z$ and assume $d\mid n$. There exists a unique subgroup of $G$ of order $d$. The mod $n$ reduction of $n/d\in \Bbb Z$ generates a $d$ element subgroup of $\Bbb Z/n\Bbb Z$, ...
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Not Abelian group G with Z(G) that contains only two elements? [on hold]

Is there an example of which is not Abelian group G, and Z(G) contains only two elements?
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$R/Rg$ is a field iff $g\in R$ is irreducible.

Let $R$ be a PID and $g\in R$. I want to show: $R/Rg$ is a field iff $g\in R$ is irreducible. I.e. I want to show that all $a\notin Rg$ are invertible modulo $g$ iff $g$ is irreducible. So if I ...
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15 views

How to prove in $r_1p_1 +r_2p_2 =u\gcd(p_1,p_2)$, $u$ is a uniformly random polynomial.

Hypothesis: All polynomials are defined over a finite field $\mathbb{F}_p$, where $p$ is a large prime number (e.g. 128-bit prime number). Assume we have two fixed polynomials $p_1$ and $p_2$ of ...
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40 views

Motivation of indices of all subgroups of symmetric group $S_n$

In 1858 a prize question of the Acad´emie des Sciences was - What are the indices of all subgroups of symmetric group $S_n$ acting on $n$ objects ? Three submissions was submitted in 1860, no ...
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22 views

Order of product multiplication of $n$-cycles that may not commute.

Suppose that $\beta$ is a $10$-cycle. For which integers $i$ between $2$ and $10$ is $\beta^i$ also a $10$-cycle? The question I have is concerning the order of checking the powers of $\beta^i$. ...
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62 views

A test problem about algebraic integers in complex field

In a recent algebraic test, I meet this problem: Let R be the ring of algebraic integers in C, K is the field of algebraic numbers in C. Let a be an element of K such that the ring R[a] is ...
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26 views

Minimal injective resolutions isomorphism [on hold]

How can I prove that given an $A$-module $M$ two injective resolutions of $M$ are isomorphic as complexes? Thank you, have a nice day Asdrubale
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18 views

noncommutative algebra

If we let $B$ be a noncommutative Banach algebra with unit $e$, then, obviously, $xy\not=yx$, but are they related? I suspect that the spectral radii of $xy$ and $yx$ are the same, but I couldn't ...
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40 views

Koszul complex: isomorphism between $K(a_1,\ldots, a_n;A) \simeq K(a_1;A) \otimes \cdots \otimes K(a_n;A)$

Given $a_1,\dots,a_n\in A$, with $A$ a suitable ring, my algebra teacher defined the Koszul complex associated to $a_1,\dots,a_n$ with coefficients in $A$ in this way: $$K(a_1,\dots,a_n;A):=\...
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1answer
33 views

Minimal graded free resolution of the ideal $(x^3,xy^2,y^5)$

I am looking for a detailed explanation of every step of the construction of a graded free resolution of the ideal $(x^3,xy^2,y^5) \subseteq S=K[x,y]$ where $K$ is an arbitrary field. I saw several ...
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1answer
44 views

About the generators of a free group

Suppose we know that $G$ is a free group of rank $n$ and that $\{g_1,...,g_n\}$, with all the $g_i$ distinct, is a system of generators for $G$. Are we sure that it is also a $\textbf{free}$ system of ...
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1answer
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Kernel of a group action of rotation of a cube

Question: Let G be the rotation group of a cube Show that G has an action on a set of size 3. Well, if we consider axes through each opposite faces, then this set has only 3 possible axes. ...
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1answer
28 views

Bijection beteween maximal ideals

We know that if $R$ and $I$ an ideal of $R$, then there is a bijection between the prime ideals of $R$ containing $I$ and the prime ideals of $R/I$. It is given by $P\mapsto P/I$. Is it true that this ...
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2answers
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Prove that if $I$ is maximal, then $R[X]$ is a PID. [duplicate]

Let $R$ be a commutative ring with unity such that $R[X]$ is a UFD. Denote the ideal $\langle X\rangle $ by $I$. Prove that If $I$ is maximal, then $R[X]$ is a PID. If $R[X]$ is a Euclidean Domain ...
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1answer
26 views

Curiosity on theorem about generating sets

Thm. (Fraleigh, 7.6) If G is a group and $a_i$ $\in$G for $i \in I$,then the subgroup H of G generated by {$a_i |\ i \in I$} has as elements precisely those elements of G that are finite products ...
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Show that a transcendental simple extension has infinitely many intermediate fields. [on hold]

Show that a transcendental simple extension has infinitely many intermediate fields. I've been working on this for awhile but can't figure it out. Help would be greatly appreciated, Thanks!
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3answers
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Show that $\mathbb{Q}(b)$ where $b=2^{1/3}+\zeta_3$ is equal to $\mathbb{Q}(2^{1/3}, \zeta_3)$ [on hold]

Show that $\mathbb{Q}(b)$ where $b=2^{1/3}+\zeta_3$ and $\zeta_3$ is a third root of unity is equal to $\mathbb{Q}(2^{1/3}, \zeta_3)$? I am not sure how to get $2^{1/3}$ and $\zeta_3$ from $b$... ...
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1answer
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Prove that up-to isomorphism there are two integral domains of order $p^2$.

Prove that up-to isomorphism there is exactly one integral domain of order $p^2$ . Does there exist only two non-commutative rings of order $p^2$ upto isomorphism? We know that any group of ...
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2answers
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Let $E=\mathbb{Q}(2^{1/3})$. What is the normal closure of $E/E$?

Let $E=\mathbb{Q}(2^{1/3})$. What is the normal closure of $E/E$? My thought is the $A(2^{1/3})$ where $A$ is an algebraic closure of $\mathbb{Q}$. But I am not sure whether it is correct and why... ...
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1answer
33 views

Functions on Finitely-Generated, Nilpotent Free, k-Algebras Determined by Values on Closed Points

I am working (slowly and with much labor) through Vakil's Algebraic Geometry and came upon this problem. Suppose $k$ is an algebraically closed field, and $A = k[x_1,... ,x_n]/I$ is a finitely ...
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1answer
22 views

Let f in F[x] be irreducible, and let E/F be Galois. Factor f in E[x] to get lower degrees.

Let $f \in F[x]$ be irreducible, and let $E/F$ be Galois. Then $f$ might factor in $E[x]$ into irreducible factors of smaller degree. Show that all of these have the same degree. I have a rough idea....
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1answer
92 views

Is it possible to endow $\text{GL}_2(\Bbb R)$ with a ring structure?

My question is the following: Is it possible to find a binary operation $*$, seen as an addition, such that $(\text{GL}_2(\Bbb R),*,\cdot)$ has a ring structure (not necessarily with a unit)? [We ...
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0answers
17 views

Example for Jacobson density theorem

I'm reading through Lang's algebra. Lang gives the Jacobson density theorem in the following way: Let $R$ be a ring (with unity) and $E$ a semisimple $R$-module. Let $R' = \operatorname{End}_R(E)...
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Show that if $H$ is a subgroup of $S_n$, then either every member of $H$

Show that if $H$ is a subgroup of $S_n$ the symmetric group of order $n$, then either every member of $H$ is an even permutation or exactly half of the members are even. I can see that if $a,b$ are ...
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1answer
32 views

Coproduct of graded Algebras

Can someone give me a construction of the coproduct of two (not necessarily commutative) GRADED algebras $A,B$ over a commutative ring $k$? I haven't found it on the internet. Thank you!
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Inclusion-exclusion principle for multisets

Lets say I want to count the number of monic polynomials of degree $d$ in $\mathbb{F}_p[X]$ that have no roots in $\mathbb{F}_p$. Fix a $1 \leq k \leq d$ and choose $k$ distinct elements of $\mathbb{F}...
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2answers
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Definition of a simple ring

I'm reading through Lang's Algebra. Lang defines a simple ring as a semisimple ring that has only one isomorphism class of simple left ideals. On the other side, Wikipedia says that a simple ring is ...
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Books on Group Theory.

I am looking for a book/note that has a good collection of advance results,theorems on finite group theory. By advance, I mean theorems after Sylow,Lagranges(I am considering theorems of Sylow,...
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1answer
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Are these algebraic properties controversial? [on hold]

I wonder whether the following properties of an algebraic system controversial or are making damage to the algebraic system rendering it useless. We have two elements, say $w_1$ and $w_2$ with the ...
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Is the assignment of a root system to a semisimple Lie algebra functorial?

As described here, we have a category of root systems, where a morphism from a root system $\Phi$ in a Euclidean space $E$ to a root system $\Phi'$ in $E'$ is given by a linear map $f: E \to E'$ such ...
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Left- and right-sided principal ideals have same index?

One fact about the Lipschitz integers (quaternions of the form $a + bi + cj + dk$ where $a, b, c, d$ are integers) is that the left-sided ideal generated by any element $Q$ has the same index in the ...
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1answer
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Converse statement related to primitive element theorem

The classical proof of the primitive element theorem (over $\mathbb Q$) implies the stronger result that if $\alpha,\beta$ are two algebraic numbers over $\mathbb Q$, then ${\mathbb Q}(\alpha+t\beta)={...
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1answer
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Group with Elements of Order 2 [on hold]

How can I prove that if a group, all the elements are from the order of $2$, then is isomorphic to $Z_2+Z_2+Z_2+..+Z_2$.
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1answer
56 views

homogeneous polynomial roots

what is the smallest field size over which a homogeneous polynomial vanishes(all its roots). For example what is the smallest field over which the following polynomial has all its roots. How to find ...
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1answer
28 views

What is the simplest example of the tame representation type?

What is the simplest example of the tame representation type? I tried to find simple example could help me to understand the tame representation type. I know the definition of tame is like: A ...
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0answers
56 views

Spec$(R)$ a scheme of finite type over $\mathbb{C} \implies R$ is a finitely generated algebra over $\mathbb{C}$.

Suppose $(\text{Spec}(R), \tilde{R})$ is a scheme locally of finite type over $\mathbb{C}$. We want to show that $R$ is a finitely generated $\mathbb{C}$-algebra. Since $(\text{Spec}(R), \tilde{R})$ ...
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The Fundamental Semigroups of a Topological Object?

I had the following idea for a generalization of the "fundamental group" of a manifold. So the idea basically was that we can consider a manifold $O$ which has boundary $\partial O$, and instead of ...
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Calculating Galois group of $\mathbb Q(\cos \frac \pi 8)/\mathbb{Q}$

What's a quick elegant way to compute the galois group of $\mathbb Q(\cos \frac \pi 8)/\mathbb Q$? I found the minimal polynomial to be $x^4-x^2-\frac 18$ but computing things directly is just ...
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1answer
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Why is the Rees Algebra Noetherian if the underlying ring is?

Let $R$ be a commutative ring with $1$, $I \subset R$ a proper ideal. The Rees Algebra, with respect to $I$, is defined: $R[It]= \bigoplus_{n=0}^\infty I^nt^n \subseteq R[t]$. In many places I've read ...
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1answer
29 views

Algebra Generated by a set modulo relations

I have a very basic question I think, but it's something that can't find in literature. So, lots of times happen to see in a book a phrase of the form "Thus, we can define the free associative $\...
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2answers
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Is a finite inverse limit of noetherian rings noetherian?

Let $\{A_i\}$ be an inverse system of (commutative, unital) Noetherian rings with a finite index set. Is $\varprojlim A_i$ also a Noetherian ring?
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Product notation for algebras

See the definition of "Graded G-algebra" on this page: https://ncatlab.org/nlab/show/crossed+G-algebra What is meant by the notation $L_gL_h\subseteq L_{gh}$ in condition (i)? In particular the ...
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1answer
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In the ring $R = \Bbb Z[\sqrt{-3}]$, prove that $2+\sqrt{-3}$ is a prime element.

In the ring $R = \Bbb Z[\sqrt{-3}]$, prove that $2+\sqrt{-3}$ is a prime element. I know the definition of prime element and have to prove that $\forall a,b\in R, 2+\sqrt{-3}|ab \Rightarrow 2+\sqrt{-...
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1answer
22 views

Steenrod Operations an algebraic Approach

Assume that $q=p^{r}$, where $p$ is a prime either 2 or odd and $\mathbb{F}_{q}$ is a Galois field and $V$ a finite dimensional $\mathbb{F}_{q}$-vector space. Then due to Larry Smith in this http://...
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1answer
36 views

Basic application of fundamental galois theorem?

Let $E/F$ be a galois extension with group $S_n$. let $G$ be the stabilizer of $1$ and $H$ be generated by the cycle $(1,\dots ,n)$. I need to find $[E^GE^H:F],[E^G:F],[E^H:F],[E^G\cap E^H:F]$. For $[...
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1answer
46 views

Are “noncommutative resolutions” a thing?

I am interested in looking at "resolutions" of modules in noncommutative rings, making some obvious necessary modifications to the definition of a resolution. However, whenever I try to search the ...
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Why is the product of two negative numbers not negative? [duplicate]

A positive number multiplied by a positive number is positive, but negative multiplied by negative is not negative but it becomes positive. How ?