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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Is there an algorithm to compute the degree of a polynomial?

Let $f\in k[X]$ be a polynomial in one unknown over any field (or any nice enough commutative ring, I imagine - it shouldn't matter) and suppose that all we can do to understand $f$ is to evaluate it ...
6
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3answers
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G/N read as G modulo N.

In my abstract algebra course, the instructor is calling G/N (the set of left Cosets of N in G) G mod N. This has not yet been explained. Why is this the case? My immediate suspicion is some ...
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1answer
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A primitive element in a field of order $r$ is a primitive $(r-1)$st root of unity.

Could someone explain to me the following sentence? "A primitive element in a field of order $r$ is a primitive $(r-1)$st root of unity." Does this mean that for each element $x$ of a field of ...
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0answers
35 views

Prove that an $n$-dimensional non-unital algebra over a field $k$ is isomorphic to a subalgebra $\mathfrak{M}_n(k)$

Assume we have already proved this for unital algebras. Here's my book's solution: Construct the unital algebra $A^1$ [with unit $(1,0)$] as an algebra on the vector space $k\oplus A$ with the ...
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1answer
39 views

General Linear Group over the quaternions is a a topological group

How to show that General Linear Group over the quaternions is a a topological group?
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1answer
31 views

When is the tensor product commutative?

I am working with the the tensor product $-\otimes_R -$ over some noncommutative ring $R$. Is the tensor product always commutative if $R$ is commutative? If so, can the tensor product be commutative ...
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11 views

Involution on the set of all multipliers of $A$ ($A$ is a $C^*$-algebra)

Let $A$ be a $C^*$-algebra. $M(A)$ denotes the set of all multipliers of $A$, i.e. $m\in M(A)$ means that there is a map $m^*:A\to A$ such that $m(a)^*b=a^*m^*(b)$ for all $a,b\in A$. I want to know ...
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1answer
32 views

Splitting fields over $\mathbb{Q}$

Find a splitting fields over $\mathbb{Q}$ for: i)$x^4+4=(x^2-2x+2)(x^2+2x+2)$ (both factors are irreducible). The roots: $x_1=1+i,\ x_2=-(1+i)$. So the splitting field is $\mathbb{Q}(i)$, which has ...
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0answers
24 views

Subgroups of $\mathbb{Z}_{13}^*$ using “GAP language”

How can I know the nontrivial subgroups of $\mathbb{Z}_{13}^*$ (with multiplication without zero) using GAP language
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2answers
28 views

The quotient of a ring by the annihilator of an ideal

Let $R$ be a commutative ring with identity and $I$ an ideal of $R$. It's true that we have an $R$-module isomorphism $$I\cong R/ann_RI,$$ where $ann_RI=\{x\in R:xr=0,\;for\;all\;r\in I\}$ is the ...
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Question concerning a minimal faithful left ideal in an artin algebra

Let $B$ be an artin algebra an suppose there is a faithful projective-injective left $B$-module. Moreover, there is a minimal faithful left ideal $Be$ for some idempotent $e\in B$. 1) What does ...
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Show that $G:=\mathbb{Z}_{13}^*$ is cyclic

I need to prove that $G:=\mathbb{Z}_{13}^*$ (without zero with multipcation)is cyclic My attempt: I tried to check each element in $G$ if it is a generator or not: $$ \begin{align} &1^1=1\mod ...
1
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1answer
35 views

Fundamental Thm of Finite Abelian Groups proof

Let $G$ be a finite Abelian group of order $p^nm$, where $p$ is a prime that does not divide $m$. Then $G=H\times K$, where $H=\{x\in G|x^{p^n}=e\}$ and $K=\{x\in G|x^m=e\}$. Moreover,$|H|=p^n$. I ...
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1answer
31 views

Every irreducible polynomial f over perfect field F is separable

Every irreducible polynomial f over perfect field F is separable. Can you check my proof? Let f is inseparable. So we have $f=\sum_i h_ix^i$ and $f^p=\sum_i h_i^px^{ip}$ Now I use Frobenius mapping ...
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2answers
20 views

Characteristic is positive and exist polynomial with $g(x^p)=f$

$F$ is a field. $f \in F[X]$ is inseparable and irreducible. Show that characteristic p of F is positive and there exists $g$ with $g(x^p)=f$. We know that f is inseparable so $gcd(f,f')\neq 1$, so ...
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0answers
81 views

Why is $\sum a_i \exp(b_i)$ always equal to $0$?

Let $z$ be complex. Let $a_i,b_i$ be polynomials of $z$ with real coefficients. Also the $a_i$ are non-zero and the non-constant parts of the polynomials $b_i$ are distinct. (*) Let $j > 1$. ...
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0answers
49 views

Does such a Galois extension exist?

Let $K = \mathbb{Q}(\sqrt{-3})$, an imaginary quadratic field. Does there exist a finite Galois extension $L/\mathbb{Q}$ which contains $K$ such that $Gal(L/\mathbb{Q})$ is isomorphic to $S_3$? Here ...
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1answer
29 views

Determining the center of the p-Sylow subgroup of $S_p $

My Algebra book says without proof that the center of the p-Sylow subgroup (we will call it P) of $S_p$ is the subgroup itself. Now I can understand that P will be a subset of its center as it is of ...
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2answers
19 views

Find isomorphism for an operation

I was trying to solve this problem, but am having trouble seeing why it is an isomorphism. To map from R* to G, I think that the phi function would be Phi(x)=x/2 but that doesn't work. This phi ...
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3answers
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What does it mean to evaluate a polynomial at a linear map, e.g. $p(T)$ where $p(x)$ is a polynomial?

I'm learning about the Cayley-Hamilton Theorem and the Primary Decomposition Theorem, and have trouble understanding what the notations mean. What does $\chi_T(T)$ mean? ($\chi_T(x)$ is the ...
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0answers
30 views

Direct sum notation

I was reading the direct sum of the groups and the index notation looks little bit strange for me. Group $G = \bigoplus_{\alpha < \beta}\mathbb Zx_{\alpha},$ where $\beta$ be an ordinal. Is this ...
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2answers
33 views

State a reason the given function is not a homomorphism

$f:\Bbb R \rightarrow \Bbb R$ and $f(x)=\sqrt x$ For $\forall x\lt0\in\Bbb R$, $f(x)=\sqrt x\in\Bbb C\notin\Bbb R$ Does my answer make sense, or should I elaborate with words?
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Eigenspaces of $Y$.

Given an even dimensional real vector space and a complex structure $Y$, why is its complexification the direct sum of $\text{ker}(X + iY)$ and $\text{ker}(X - iY)$?
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8answers
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What does it even mean to say 'preserve structure'? [duplicate]

Could somebody give a concrete example of a group structure being preserved in a isomorphism, et cetera? I always hear this 'preserve structure' thing. Ok, could somebody give me a rigorous definition ...
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3answers
36 views

Showing that a subgroup of an abelian group is normal—is this sufficient?

When asked to show that a subgroup $H$ of the abelian group $G$ is normal, does it suffice to say: first, $H$ is a subgroup, so it contains the identity element of $G$ and inverses $h^\prime$ for ...
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1answer
26 views

Latin square property sufficient?

So I know that for any group table, Every row must contain distinct group elements and the same holds for every column for a group table. And this property is called the Latin Square property. ...
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24 views

A subgroup of special linear group

Does anybody know if the subgroup of diagonal and antidiagonal matrices of $SL(n,F)$ has been given a particular name? By $SL(n,F)$ I mean $n \times n$ matrices over a field $F$ with determinant 1. ...
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2answers
34 views

Equivalent condition for split exact sequence

the question is in Module Theory, Let $M,N,\&\ L$ be any R-modules. Then , for any short exact sequence $0\longrightarrow N\overset{f}{\longrightarrow} M\overset{g}\longrightarrow L ...
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1answer
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Comparing two statements of Chinese Remainder Theorem (Sun-Ze Theorem)

Wikipedia states the Chinese Remainder Theorem as follows: Suppose $n_1, \dots, n_k$ are positive integers which are pairwise coprime. Then for any given sequence of integers $a_1, \dots, a_k$, ...
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Is the given subset a subspace of the given vector space?

The set of all polynomials of degree greater than 3 together with the zero polynomial in the vector space P of all polynomials with coefficients in $\Bbb R$. Let $S$ be the set of all polynomials ...
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2answers
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Prove that the free group generated by two elements is a coproduct of the integers by itself in Grp

That's a problem from Algebra: Chapter $0$ by P. Aluffi, p.78, ex.5.6. One needs to prove that the group $F(\{x,y\})$ is a coproduct $\mathbb Z*\mathbb Z$ of $\mathbb Z$ by itself in the category ...
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1answer
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Is every field a Krull domain?

Background: A Krull domain is an integral domain $A$ with a family $(v_i)$ of valuations on the field of fractions $K$ for $A$ satisfying the following conditions: Each $v_i$ is discrete. The ...
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1answer
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Construction of free abelian group from free group

I am reading Fraleigh's Abstract Albebra recently, and I cannot prove a statement about free abelian group: Let $F[A]$ be a free group generated by set $A$ and $C$ is the commutator subgroup of ...
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Is there an example such that $\text{rank}(A^t)\neq \text{rank}(A)$?

Let $R$ be a commutative Noetherian ring and $A\in M_{n\times m}(R)$. If $R$ is a field, then $rank(A^t)=rank(A)$. However, in general Noetherian rings, does $rank(A^t)=rank(A)$ hold? Since the ...
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2answers
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Maximal closed subgroups in algebraic groups

Let $G \leq GL(V)$ be an affine algebraic group, over an algebraically closed field. Say that $M$ is a proper subgroup of $G$ which is maximal among the closed proper subgroups of $G$. Does $M$ have ...
3
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2answers
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Flat extension of noetherian rings and formal power series

Let $A \to B$ be a flat homomorphism of Noetherian rings. Is it true that it induces a flat homomorphism of formal power series $A[[x]] \to B[[x]]$?
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1answer
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If $F(\alpha)=F(\beta)$, must $\alpha$ and $\beta$ have the same minimal polynomial?

Let's consider a field $F$ and $\alpha,\beta\in\overline{F}-F$ (where $\overline{F}$ is an algebraic closure). If $F(\alpha)=F(\beta)$, is it true that $\alpha$, $\beta$ have the same minimal ...
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2answers
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Definition of algebraically closed field.

A field F is algebraically closed if every non constant polynomial in F[x] has a root in F. Is this the right definition? I am wondering if only one root in F and the rest of the roots not in F can ...
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Generators of the group of invertible elements of the ring $\mathbb{Z}_{14}$—are they multiplicative or additive?

When I was asked to find the generators of the group of invertible elements of the ring $\mathbb{Z}_{14}$, which are denoted as $\phi(14)$, I did not realize whether the generators are multiplicative ...
3
votes
1answer
44 views

Why is the commutator group a subgroup?

I am in Intro to Algebra, and have a question regarding the commutator subgroup. I am a bit confused with the premise, though, with how the set is a subgroup in the first place. Let $C$ be the set of ...
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1answer
17 views

Number of equivalence Relations containing $(1,2)$

Find the number of equivalence Relations on the Set $A=\{1,2,3 \}$ which contains the Element $(1,2)$. My Try: Since $(1,2)$ is to be included, so is $(2,1)$ since the Relation should be Symmetric ...
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How to compute $\mathbb Z_n \times \mathbb Z^*_m$? [on hold]

How to compute $\mathbb Z_n \times\mathbb Z_m^*$? (Here $\mathbb Z^*_m$ is the unit group mod $m$ and $(m,n)=1$.) In the paper Multiplicative properties of sets of residues it is said that by ...
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Looking for help to understand example of Group

I am looking for someone to help me to understand what is going on in the following example, from Hersteins "Topics in Algebra". It says, Let $G$ be the set of all $2*2$ matrices $$\begin{pmatrix} ...
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1answer
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Unique factorization domain: $\mathbb{Z}_{n}[x]$

How to determine all $n\in\mathbb{N}$ such that $\mathbb{Z}_{n}[x]$ is a unique factorization domain? I am guessing that this would be true for all primes, since $\mathbb{Z}_n$ is a UFD when $n$ is ...
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1answer
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$F[[T]] \times F[[1/T]]$, fundamental domain.

Let $p$ be a prime number. Here is a link which shows how to see that $$(\mathbb{F}_p((T)) \times \mathbb{F}_p((1/T)))/\mathbb{F}_p[T, 1/T]$$is compact using an adelic result. (Here $\mathbb{F}_p[T, ...
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+50

Subring of $\mathcal O(\mathbb C)$

Let $\mathfrak A \subset \mathcal O(\mathbb C)$ be the subring generated by the nowhere zero analytic functions $f: \mathbb C \to \mathbb C$. Does we have a precise description of $\mathfrak A$ ? Is ...
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$y^2 = x^3 - 26$, exist ideal satisfying conditions?

For the solution $(x, y) = (3, 1)$ of $y^2 = x^3 - 26$, does there necessarily exist an ideal $I$ of the integer ring $\mathbb{Z}[\sqrt{-26}]$ of $\mathbb{Q}(\sqrt{-26})$ such that $(y + ...
3
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1answer
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If $G$ is a group of order $2^nm$, where $m$ is odd and $(m-1)!<2^n$, show that $G$ is not simple.

If $G$ is a group of order $2^nm$, where $m$ is odd and $(m-1)!<2^n$, show that $G$ is not simple. I started out by trying to prove this using the Sylow theorem, but it led nowhere. I was able to ...
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1answer
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Ideal in $\mathcal O(\mathbb C)$

Let $\mathfrak {I}$ the ideal generated by all the holomorphic functions which are never zero. Question : is $\mathfrak {I} = \mathcal O(\mathbb C)$ ?
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From group isomorphisms to algebra isomorphisms

Let $A$ be an algebra and let $A^{\ast}$ be the subset of units (that is, invertible elements) of $A$. Then $A^{\ast}$ is a group under the multiplication of $A$. Let $f^{\ast}:A^{\ast}\to A^{\ast}$ ...