# All questions with tag [math: abstract-algebra]

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## Problem connecting Topology and Algebra via Analysis

Let $C(X):=$ Set of all complex/real valued continuous functions. If $X$ is compact then all the maximal ideals in the ring $C(X)$ is of the form $M_{x}=\{f\in C(X): f(x)=0\}$ for some $x\in X$. Is it true that: If all the maximal ideals in the ring $C(X)$ is of the form $M_{x}=\{f\in C(X): f(x)=0\}$ for some $x\in X$ then $X$ is compact.
2022-07-25 20:47:10
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## On Constructions by Marked Straightedge and Compass

Pierpont proved that a regular $n$-gon is constructible by (singly) marked straightedge and compass if and only if $n = k \, p_1 \cdots p_{s}$, where $k = 2^{a_1} 3^{a_2}$ for $a_i \geq 0$ and $p_i = 2^{b_1} 3^{b_2} + 1 &gt; 3$ is prime with $b_i \geq 0$. It has been known since the time of Archimedes that a marked straightedge allows for an...
2022-07-25 20:47:10
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## What exactly is an $R$-module?

From : "If $R$ is commutative, then left $R$-modules are the same as right $R$-modules and are simply called $R$-modules." The definition of left $R$-module: $M$ is a left $R$-module if $M$ is an abelian group and $R$ a ring acting on $M$ such that (i) $r(m_1 + m_2) = rm_1 + rm_2$ (ii) $(r_1 + r_2 ) m = r_1 m + r_2 m$ (iii) $1m = m$ (...
2022-07-25 20:46:44
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This is a newbie is inquiry on what specifically is a tensor item, in nonprofessionals is term, for a newbie that has actually simply found out standard team concept and also standard ring concept. I do recognize from wikipedia that in many cases, the tensor item is an external item, which takes 2 vectors, claim $\textbf{u}$ and also $\textbf{v... 2022-07-25 20:45:20 0 ## Solubility of a Galois Group going over some past papers with no answers and would like a bit of help if possible.. I've shown that for p a prime number then$x^p-1 \in K[x]$is abelian where K is a subfield of$\mathbb{C}$. I've now been asked to show that the Galois group of$x^p-a$over K is soluble with$a \neq 0 \in K$. I know any abelian group A is soluble, since${...
2022-07-25 20:43:41
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## Relating $\operatorname{lcm}$ and $\gcd$

I would appreciate help to show this equality is valid: $\operatorname{lcm} (u, v) = \gcd (u^{- 1}, v^{- 1})^{- 1}$, where $u, v$ are elements of a field of fractions. In the text it is stated that lcm is: there is an element $m$ in K for which $u| x$ and $v| x$ is equivalent to $m| x$ It goes on to say sending $t$ to $t^{- 1}$ in K reverses di...
2022-07-25 20:41:44
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I'm trying to figure out what is the correct way to compute a check matrix for a binary Goppa code. So far (searching through the publications) I've found more than one possibility to do that, and I'm not sure how all those are related. Suppose I have a Goppa code with irreducible Goppa polynomial $g(x)$ of degree $t$ over $GF(2^m) \simeq F[x]/F... 2022-07-25 20:40:56 1 ## The divided polynomial algebra over a field Let$\Gamma_R[\alpha]$denote the divided polynomial algebra over$R$; that is, the quotient of the free$R$-algebra$R\langle \alpha_1,\alpha_2,\cdots \rangle$by the relations $$\alpha_n \cdot \alpha_m = \binom{n+m}{n}\alpha_{n+m},$$ with$t_0=1$. I am particularly interested in the case where$R = \mathbb{F}_p$. The claim is that $$\Gamma_{... 2022-07-25 20:22:55 1 ## k-basis of a quotient of ideals in polynomial ring Let k be an area and also take into consideration the excellent I=(x,y) \subset k[x,y]. Am I deal with in claiming that (x,y)/(x,y)^{2} is created as a k - vector room by the class of x and also y? 2022-07-25 20:21:35 0 ## Why is such an ideal ambiguous? Suppose I have an R-ideal I with$$I=(1-\zeta)^n XR$$with R=\mathbb{Z}[\zeta], \zeta a primitive p-th root, X an ideal in the integral closure of \mathbb{Z} in \mathbb{Q}[\zeta + \zeta^{-1}] (would it be \mathbb{Z}[\zeta + \zeta^{-1}] ?) and the exponent n is 0 or 1. Why is such an ideal ambiguous? I can calculate it ju... 2022-07-25 20:20:36 1 ## Subgroups of finite solvable groups. Solvable? I am attempting to prove that, given a non-trivial normal subgroup N of a finite group G, we have that G is solvable iff both N, G/N are solvable. I was able to show that if N,G/N are solvable, then G is; also, that if G is solvable, then G/N is. I am stuck showing that N must be solvable if G is. It seems intuitive that an... 2022-07-25 20:20:25 1 ## Is inclusion of a prime ideal into a different prime ideal possible? Let A be a commutative ring with identification. Allow p, q be 2 distinctive prime perfects. Is it feasible that p \subseteq q? 2022-07-25 20:19:15 1 ## Free Group Generated By Image Let S be any set and let f:S \rightarrow F denote the free group on S. By definition, this means that for any group X and any function g:S \rightarrow X there exists a unique homomorphism h:F \rightarrow X such that h \circ f = g. How can I show that f(S) generates F? By "generate F" I mean that the intersection of ... 2022-07-25 20:18:31 1 ## Is the image of a tensor product equal to the tensor product of the images? Let S be a commutative ring with unity, and also allow A,B,A',B' be S - components. If \phi:A\rightarrow A' and also \psi:B\rightarrow B' are S - component homomorphisms, is it real that$$\operatorname{im}(\phi\otimes\psi)=\operatorname{im}(\phi)\otimes_S \operatorname{im} (\psi)?$$2022-07-25 20:18:05 2 ## Proving that the length of the sum and intersection of two finite length modules is finite. I am attempting to address the adhering to question: Let M_1, M_2 \subset M be limited size submodules of a component M (where M need not be of limited size). Show that \ M_1 + M_2 = \{x_1 + x_2 \in M \, | \, x_i \in M_i\} and also M_1 \cap M_2 have limited size which l(M_1) +l(M_2) = l(M_1 \cap M_2) + l(M_1 + M_2). I am actually ... 2022-07-25 20:15:56 1 ## Intuition surrounding units in R[x] My lecture notes state that an 'easy' result is If R is an integral domain then an irreducible element of R remains irreducible in R[x], and the units in R and in R[x] are the same. I can't seem to get my head around why this is the case, and what a unit in R[x] means intuitively because I don't see how the units can be the same if... 2022-07-25 20:00:17 1 ## The definition of a unique factorisation domain Wikipedia offers the definition of a Unique Factorisation Domain as one where every component "can be created as an item of prime components (or irreducible components) " which recommends that in a UFD prime and also irreducible components coincide. Nonetheless, I assumed that just in a PID were prime and also irreducible components t... 2022-07-25 19:59:08 1 ## Proving that when \nu(x) = \nu(y) in the Gaussian integers they are associates I assume I'm missing out on something below yet my lecture keeps in mind simply appear to state that 'plainly' for x and also y in the Gaussian integers (components of \mathbb{Z}[i], a Euclidean domain name), if \nu(x) = \nu(y) after that plainly x and also y are affiliates. Am I missing out on something below or is this noticeable? 2022-07-25 19:59:00 2 ## Let N be a submodule of the module M. Suppose M/N and N are semi-simple. Does it follow that M is semi-simple? Let N be a submodule of the module M. Suppose M/N and N are semi-simple. Does it follow that M is semi-simple? I think the answer is yes but I am not sure how to prove it. Any help would be appreciated. 2022-07-25 19:51:57 1 ## Generators for PSL(2,\mathbb{Z}) with a specific property Does there exist generators S and also T for the modular team \Gamma=PSL(2,\mathbb{Z}) with the adhering to property:$$S+S^{-1}+T+T^{-1}=0$$Here is a candidate:$$S=\left[\begin{array}{cc} -1 &amp; 0 \\ 1 &amp; -1 \\ \end{array}\right], \,T=\left[\begin{array}{cc} 1 &amp; 1 \\ 0 &amp; 1 \\ \end{array}\right]$\$ Just no...
2022-07-25 19:44:20