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Questions tagged [abstract-algebra]

Abstract algebra is the study of algebraic objects. Some of the more common algebraic objects are groups, rings, fields, vector spaces, and modules, among other topics. Whenever you use this tag, please also include related tags like group-theory, ring-theory, modules, etc., in order to clarify which topic of abstract algebra is most related to your question and also to help other users find similar questions through the search engine.

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2answers
29 views

GCDs for the polynomial ring over a Galois field.

You can find many examples of computing the inverse of an element inside a Galois field. (For example here) What happens if we look at the polynomial ring over a Galois field and would like to ...
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1answer
41 views

Irreducibility of the following Polynomial over $\mathbb{Q}$

Take the polynomial $x^4 + 10x^2 + 1$. Is this irreducible over $\mathbb{Q}$? If so, what is the best way to show this? I know it can be rewritten: $$ (x^2 + 5)^2 -24 $$ Which can be simplified over ...
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1answer
14 views

Proving that two subsets of a set of bijections are either equal or disjoint

$X$ is a set and $X!$ is the set of bijections from $X$ to $X$. Let $A(m)$ be the set defined as $A(m)=\{f^x(m)\mid x\in\mathbb Z\}$ for some $f$ in $X!$ and some $m$ in $X$. Now, how do I prove that ...
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0answers
66 views

Pythagorean closure

Im reading the book "Galois Theory" by Ian Stewart $(4$th Edition$)$. Here the author defines the Pythagorean closure as follows: Definition:The Pythagorean closure $\mathbb{Q}^{PY}$ of $\mathbb{Q}...
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2answers
42 views

Is every ideal in $K[x, y]$ of finite codimension necessarily prime?

I'm trying to answer the following question: Suppose that $R$ is an integral domain containing a field $K$. Then we may view $R$ as a $K$-vector space. Show that if R is finite dimensional as a K-...
3
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0answers
53 views

Factoring $x^n - 1$ and minimal polynomials

Let $gcd(n,q) = 1$ I'm trying to get to grips with factorising the polynomial $x^n - 1$ over $\mathbb{F}_q$. Firstly, it is a good idea to find an extension field containing all the roots of $x^n - 1$...
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1answer
44 views

Affine $n$-space over a scheme

In an exercise of Eisenbud-Harris The Geometry of Schemes, they ask to prove the following: Let $S$ be any scheme. Let $\mathbb{A}_{\mathbb{Z}}^n = \mathrm{Spec}\mathbb{Z}[x_1, \dots , x_n]$ be ...
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2answers
15 views

skew Schur functions $C^{\lambda}_{\mu, \nu}$

When working with skew Schur functions, they can be defined as follows. Let $C^{\lambda}_{\mu, \nu}$ be the integers such that $$s_{\mu}s_{\nu}=\sum_{\lambda} C^{\lambda}_{\mu, \nu} s_{\lambda}$$ ...
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3answers
58 views

$\Lambda = \varprojlim\Lambda_n$ (ring of symmetric functions)

This question is related to this other question. When understanding how it is defined the ring of symmetric functions, I can not see why is so much important to take the inverse limit in the category ...
4
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1answer
47 views

Find two odd primes for which $(p-1)!≡-1\mod p^2$ where $p \le13$?

Except brute force is there some way to solve this ? One way we can solve it by using Wilson's theorem but I was not able to proceed much.
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0answers
19 views

Showing two series are not the same.

I want to show that the following two composition series are not the same: $D_{8}\triangleright \left \langle s,r^{2} \right \rangle \triangleright \left \langle s \right \rangle \triangleright (1)$ ...
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1answer
41 views

Solve the equations $x^2= x,~x^2=1$ and $x^{32}=1$ on $\mathbb{Z}_{128}$

On ring $\mathbb{Z}_{128}$, solve each of the following equations: $$(i)~x^2= x,~~~~~~(ii)~x^2=1,~~~~~~(iii)~x^{32}=1.$$ Attempt. Some thoughts. (i) Clearly $x=0,1$ are solutions. Let $x \in \mathbb{...
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1answer
35 views

Showing this is an automorphism

Let $R_n = F_q[x]/\langle x^n - 1 \rangle$, where $F_q[x]$ is a finite field. Consider $\mu_a$ which acts on $R_n$ like so; $f(x) \mu_a \equiv f(x^a) \bmod (x^n - 1)$ for $f(x) \in R_n$. Is this an ...
2
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0answers
41 views

Elementary proof that centre of finite $p$ group is not trivial [duplicate]

Prove that the centre of finite $p$ group is not trivial. I have found on many links proofs of this property, but all of them use the "Class Equation". I would like to know if there is a proof which ...
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2answers
42 views

Find quantity of elements in group with given order

Let $G = ( \mathbb { Z } / 133 \mathbb { Z } ) ^ { \times }$ be the group of units of the ring $\mathbb { Z } / 133 \mathbb { Z }$ . Find the number of elements of $G$ of order $9 .$ 133 cannot ...
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0answers
26 views

Uncountable applications of an operator?

To simplify matters, assume we have a commutative group $(X,\cdot,1)$ with uncountable $X$. For commutative groups, applications of elements $x_i\in X$ don’t care about order, and we can simply count ...
3
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1answer
43 views

How to prove directly sum of non zero divisor and nilpotent is again non zero divisor?

How to prove directly sum of non zero divisor and nilpotent is again non zero divisor? I know that it can be easily proved by extending ring to ring of fraction So that I have a unit as that non zero ...
0
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1answer
27 views

Let $R$ be a domain and let $M \subset R$ be a maximal ideal. Let $K$ be the quotient field of $R$

Let $R$ be a domain and let $M \subset R$ be a maximal ideal. Let $K$ be the quotient field of $R$. Let $T = \{\frac{r}{s} , s\notin M\} \subseteq K$, and $M_1 = \{\frac{m}{s},s \notin M, m \in M\} \...
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2answers
47 views

coprime ideals in $K[X]$

If $K$ is a field, $A=K[X]$, take $m,n \in K$ such that $m \ne n$. Prove that the ideals $I=(X-m)$ and $J=(X-n)$ are coprime. I know the regular definition of coprime. But here, should we prove $I + ...
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0answers
20 views

Find specific elements with a given order in quotient group

How can I deal with the quotient group generated by several element? And how can I find the element with a given order.
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1answer
36 views

$M$ and $N$ flat, then $M\otimes N$ flat

I want to show that if $M$ and $N$ are flat $R$-modules, then $M\otimes_R N$ is flat. By flat we mean that if $0\to A\to B$ is exact, then $0\to A\otimes_RM\to B\otimes_RM$ is exact. I am assuming ...
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1answer
26 views

definition of affine polynomial [closed]

I'm reading a paper and it is written inside it "affine polynomial" but I don't know this definition, and couldn't find it on the web. Could you please help me if you know it?
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1answer
24 views

Ring with additional constraints on distributivity

I have a ring with the exception that the distributive property $$a*(b+c)\neq a*b+a*c$$ doesn't always hold. However it will hold under certain additional constraints for example $b>0,c>0$. So ...
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1answer
33 views

bijection between a set and a set of functions

Question: For any function $f : A → B$ define an explicit isomorphism between A and the graph $Γ_f ⊂ A×B$ the subset defined by the property that for each $a ∈ A$ there is exactly one pair $(a, b) ∈ ...
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0answers
28 views

Group Isomorphisms with $D_k$ [closed]

Show that $D_k$ is isomorphic to the group generated by \begin{bmatrix}0&1\\1&0\end{bmatrix}and\begin{bmatrix}ζ&0\\0&ζ^{-1}\end{bmatrix}under matrix multiplication, where $ζ = exp(\...
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0answers
51 views

not understand functor of points definition

This is definition of scheme functor in Mumford, Oda, Algebraic Geometry II Chpt 1. Let $F$ be a covariant functor from Ring category to set category s.t. for any ring $R$, $Spec(R)=\cup_iD(f_i)$, $F(...
1
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2answers
16 views

Arithmetic functions on short exact sequences

Consider a short exact sequence $0\to A\to B\to C\to 0$ of abelian groups or modules over a fixed ring $R$ or complexes of modules over $R$. Denote abelian group category or module category as $\...
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0answers
17 views

Strong Approximation theorems in Function Fields versus Groups

In the context of global function fields, the strong approximation theorem can be stated as follows: Let $F$ be a global function field, and $P_1,P_2,\dots,P_r$ be a finite set of places of $F$ (with ...
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1answer
27 views

When do $M_n, GL_n, SL_n$ commute with direct products?

We know, for example, that $SL_2(\mathbb Z/n\mathbb Z)\cong \oplus_{p\mid n}SL_2(\mathbb Z/p^{e_p}\mathbb Z)$. To what extend does this hold in general? That is, if we're given, say, some commutative ...
2
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1answer
31 views

Construct a group that has exactly 5 elements of order 4.

Construct a group that has exactly 5 elements of order 4. I wonder if it is possible. I tried $U(8)$ but it has $\{[1], [3], [5],[7]\}$ as elements which has order $4$ but it has only $4$ elements. ...
1
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1answer
39 views

For a ring $R,$ there is no injective map from $R^m \hookrightarrow R^n$ for $m > n.$ (verification)

Suppose $R$ is a ring and we have an $R$-module homomorphism of free modules $R^{n + 1} \hookrightarrow R^n.$ Then take a submodule of $R^{n + 1}$ that is isomorphic to $R.$ Say for any $r \in R,$ we ...
1
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1answer
23 views

Decomposition of $K$ scheme points of fiber product

Let $K,X,Y$ be $S-$schemes(i.e. $K\to S, X\to S, Y\to S$ are given and fixed.) Consider $Z=X\times_S Y$ and $\phi: Hom(K,Z)\to Hom(K/S,X/S)\times Hom(K/S,Y/S)$ where $Hom(K/S,X/S)$ is the set of ...
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1answer
39 views

Find the values of $a$ so that $A$ is positive definite (p.d.)

Let $A=(1-a)I_n + a J_n$. Find the values of $a$ so the matrix is p.d. Note: $I_n$ is the identity matrix and $J_n$ is the $1's$ matrix. I know that $A$ is p.d. iff $λ_i >0$ so, I need to choose $...
2
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1answer
76 views

Prove that $2^n$ is not a polynomial in an algebraic way.

There is no polynomial $P \in \mathbb{R}[X]$ such that $P(n) = 2^n$ for all $n\in\mathbb{N}$. I already know the analysis way to prove this using $\lim$, derivations or Taylor series. But this is an ...
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1answer
28 views

Given the operation X$*$Y = $X^c \cap Y^c$ proof that G={P(A),$*$} it is not group [closed]

Given the set A={0,1,2,3,4} and the operation X$*$Y = $X^c \cap Y^c$ proof that G={P(A),$*$} it is not group
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0answers
27 views

Question about commutativity of conjugacy classes

I have an exercise in group theory which asks to show that conjugacy classes commute even though the group is not abelian, i.e. that $C_1C_2 = C_2C_1$ for any classes $C_1$ and $C_2$. The proof given ...
1
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1answer
38 views

What is the tensor product $\mathcal{O}_K \otimes_\mathbb{Q} \mathbb{R}$?

Let $m \in \mathbb{N}^*$. Denote the $m$-th cyclotomic polynomial by $\Phi(x)$ and a complex primitive $m$-th root of unity by $\omega$. Let $K = \mathbb{Q}[x]/\langle \Phi(x) \rangle$ (which is ...
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0answers
34 views

Modular Polynomial Arithmetic in Schoof's Algorithm

I've been trying to implement Schoof's Algorithm, and I understand it except for one part. Near the bottom of page 7 of this paper is where my issue is: http://www-users.math.umn.edu/~musiker/schoof....
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1answer
84 views

When is $A\rtimes_{\phi_1} B \cong A\rtimes_{\phi_2} B$?

Suppose $1\to K \stackrel{m}{\rightarrow} G \stackrel{f}{\rightarrow} H \to 1$ is short exact sequence of groups. The followings are equivalent: $(1)\ G\cong K \times H;$ $(2)$ The sequence right ...
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0answers
38 views

Axes of a free module over a PID (2)

Let $R^n$ be a finitely generated free module of rank $n > 0$ over a principal ideal domain. I am trying to prove that for every non-zero element $a$ of $R^n$ there is a basis such that $a$ ...
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1answer
52 views

Suggestion of books

Respected all members, I want to read a book(Abstract algebra, Real analysis, Topology) in which each mathematical concepts is related with real life. Book which is totally based on application. I ...
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1answer
27 views

Can every positive root of a Coxeter group be written as a simple root and a positive root?

Can every positive root of a Coxeter group be written as a simple root and a positive root? I think that this is possible. For example, in type $B_2$, the set of positive roots are $\alpha_1, \alpha_2,...
3
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2answers
59 views

Show that $2018^{6k+2}$ can't be written as $a^3+b^3+c^3$, where $a,b,c\in \mathbb {N} $.

Show that $2018^{6k+2}$ can't be written as $a^3+b^3+c^3$, where $a,b,c\in \mathbb {N} $. I tried to prove that this is not possible by congruences modulo 3, 5, 11, 13, 17 but it doesn't work.
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2answers
63 views

$x^3 - 3x - 1$ irreducible in $\mathbb Z[x]$ by Gauss Lemma

In Dummit & Foote, they claim this can be shown to be irreducible by Gauss Lemma and applying it to show it has no rational root. But this doesn't make sense to me since Gauss Lemma says: ...
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2answers
52 views
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1answer
34 views

How to prove that multiplicative group of integers modulo n contains only integers coprime to n?

I have this question given as exercise in my discrete math book. Since $(\mathbb Z/n \mathbb Z)^\times$ is a group, every element of it has an inverse. So, how to prove that all those integers are ...
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2answers
31 views

Finding all group morphisms [closed]

Find all the group morphisms from $(\mathbb Z, +)$ to $(\mathbb Q, +)$. I don't have any idea where to start, any help would be great!
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1answer
43 views

Suppose $H$ is a nonempty subset of a finite group $G$. How to show $H$ is a subgroup of $G$ if and only if $ab \in H$ for all $a,b \in H$.

One direction follows from the definition of subgroup. If $H$ is a subgroup of $G$, then it must be closed under the operation. However, if $ab\in H$ for all $a,b \in H$, then it is closed under the ...
1
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1answer
28 views

Simple question about flat modules

Everything I can find about this is stated in category theory language that I do not understand. If I have an exact sequence $... \rightarrow A \xrightarrow[\text{}]{\text{f}} B \xrightarrow[\text{}]{...
7
votes
1answer
96 views

Is a finite centerless metabelian group always a semidirect product of two abelian groups?

Suppose $G$ is a finite centerless metabelian group. Is it true that it is a semidirect product of two abelian groups? It does not seem true to me, but I failed to find any counterexamples. Actually, ...