All questions with tag [math: abstract-algebra]


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 17:47:10

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 $p_i = 2^{b_1} 3^{b_2} + 1 > 3$ 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 17:47:10

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 17:46:44

What exactly is a tensor product?

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 17:45:20

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 17:43:41

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 17:41:44

Proper way to create Goppa code check matrix?

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 17:40:56

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 17:22:55

$k$-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 17:21:35

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 17:20:36

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 17:20:25

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 17:19:15

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 17:18:31

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 17:18:05

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 17:15:56

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 17:00:17

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 16:59:08

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 16:59:00

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 16:51:57

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: $$S=\left[\begin{array}{cc} -1 & 0 \\ 1 & -1 \\ \end{array}\right], \,T=\left[\begin{array}{cc} 1 & 1 \\ 0 & 1 \\ \end{array}\right]$$ Just no...
2022-07-25 16:44:20