 # Abstract algebra questions and answers

Recent questions in Abstract algebra et3atissb 2022-04-23 Answered

### Is k[[x]] ever a finitely generated k[x](x) module?For k a field, the localization $k{\left[x\right]}_{\left(x\right)}$ naturally includes into k[[x]]. I can prove that if $k=\mathbb{C},$, then this inclusion is not surjective, and k[[x]] is not even finitely generated over $k{\left[x\right]}_{\left(x\right)}$, because $\mathbb{C}{\left[x\right]}_{\left(x\right)}$ corresponds to rational functions holomorphic at 0, and $\mathbb{C}\left[\left[x\right]\right]$ corresponds to germs of all functions holomorphic at 0, and so with some complex analysis you can see the finite generation is impossible. But over an arbitrary field, is this claim still true? windpipe33u 2022-04-23 Answered

### Show that the Lorentz boosts, $\left(\begin{array}{cc}\gamma & -\beta \gamma \\ -\beta \gamma & \gamma \end{array}\right)$form a one-parameter Lie group. coraletsmmh 2022-04-23 Answered

### Product of two cyclic groups is cyclic iff their orders are co-'Say you have two groups $G=⟨g⟩$ with order n and $H=⟨h⟩$ with order m. Then the product $G×H$ is a cyclic group if and only if $gcd\left(n,m\right)=1$ Zack Wise 2022-04-22 Answered

### Notation of homeomorphism from B(H) to B(K), corresponding to unitary transformation of Hilbert spacesLet U be a unitary transformation from Hilbert space H to Hilbert space K.How do you call a *-homomorphism f from B(H) to B(K), defined by $f\left(a\right)=Ua{U}^{-1}$?I'm interested both in a symbol, which can be used in formulas, and a name for it, which can be used in texts or in speech. Saige Shannon 2022-04-22 Answered

### How does cancellation work in polynomial quotient rings?$\frac{\mathbb{Z}\left[x\right]}{\left(2x-6,6x-15\right)}$Can I just automatically say that$\frac{\mathbb{Z}\left[x\right]}{\left(2x-6,6x-15\right)}\stackrel{\sim }{=}\frac{\mathbb{Z}\left[x\right]}{\left(x-3,6x-15\right)}$by just dividng the first polynomial by 2? Dereon Guzman 2022-04-21 Answered

### If the cardinality of a ring R is greater than 1, then $1\ne 0$. devojciq5o 2022-04-20 Answered

### Solving (quadratic) equations of iterated functions, such as$f\left(f\left(x\right)\right)=f\left(x\right)+x$ Jaylyn Villarreal 2022-04-17 Answered

### We want to prove that $\left({Z}_{7},\oplus \right)$ is group. I have difficulty proving associativity axiom. The solution readsLet $a\in {\mathbb{Z}}_{7},b\in {\mathbb{Z}}_{7}$ and $c\in {\mathbb{Z}}_{7}$. By Theorem 3.4.10 we only need to show$\left(a+\left(b+c\right)\right)b\text{mod}7=\left(\left(a+b\right)+c\right)b\text{mod}7.$This holds since $a+\left(b+c\right)=\left(a+b\right)+c$ for all integers a, b, and c by the associative property of the integers. Hence $\oplus$ is associative.Therorem 3.4.10. Let a and b be integers, and let m be a natural number. Then  $\left(a+b\right)modm=\left(\left(amodm\right)+\left(bmodm\right)\right)modm$ Kendal Day 2022-04-16 Answered

### Multiples of 4 as sum or difference of 2 squaresIs it true that for any $n\in \mathbb{N}$ we can have $4n={x}^{2}+{y}^{2}$ or $4n={x}^{2}-{y}^{2}$, for $x,y\in \mathbb{N}\cup \left(0\right)$?I was just working out a proof and this turns out to be true from . After that I didn't try, but I would like to see if a counterexample exists for a greater value of n. Essence Ingram 2022-04-16 Answered

### We were learning about centralizers in my abstract algebra class, and I had a thought about centralizers. Is every subgroup of a group G the centralizer of some element? And if not, how many of the subgroups are? gabolzm6d 2022-04-16 Answered

### What are all the homomorphisms between the rings ?Any homomorphism $\psi$ between the rings is completely defined by $\psi \left(1\right)$. So from$0=\phi \left(0\right)=\phi \left(18\right)=\phi \left(18\cdot 1\right)=18\cdot \phi \left(1\right)=15\cdot \phi \left(1\right)+3\cdot \phi \left(1\right)=3\cdot \phi \left(1\right)$We get that $\psi \left(1\right)$ is either 5 or 10. But how can I prove or disprove that these two are valid homomorphisms? Papierskiix5n 2022-04-15 Answered

### Let A be a discrete valuation ring, and let $a\in A$ be a non-zero element. Compute the integral closure of Santos Mooney 2022-04-14 Answered

### Let G be a finite group. Let T be an element of Aut(G) such that${T}^{2}=I$, and $xT=x⇔x=e$.Then, G is abelian. Breanna Mcclure 2022-04-14 Answered

### Is there a particular characterization of $Aut\left({\mathbb{Q}}^{3}\right)$? Bradley Barron 2022-04-14 Answered

### Find the isomorphic ring with𝟞${\mathbb{Z}}_{\mathbb{6}}\frac{x}{⟨{x}^{3}-x⟩}$(Chinese remainder thm) Rubi Riggs 2022-04-13 Answered

### Let G be a finite group with $card\left(G\right)={p}^{2}q$ with $p two ' numbers. We denote ${s}_{q}$ the number of q-Sylow subgroups of G and similarly for p. I have just shown that . Now I want to show that$\underset{S\in Sy{l}_{q}\left(G\right)}{\cup }S\setminus \left\{1\right\}={\stackrel{˙}{\cup }}_{S\in Sy{l}_{q}\left(G\right)}S\setminus \left\{1\right\}$i.e. that for  with $S\ne T$ we that $\mathrm{S}\setminus \left\{1\right\}\cap \mathrm{T}\setminus \left\{1\right\}=\varnothing$ Ansley Sparks 2022-04-13 Answered

### Jacobson radical of the ring of lower triangular $n×n$ matrices over $\mathbb{Z}$ Rosa Townsend 2022-04-13 Answered

### Isomorphism between quotient of ring and principal idealsLet R be a principal ideal domain and suppose that $u,{u}^{\prime }\in R$ are such that $\frac{R}{\left(u\right)}\stackrel{\sim }{=}\frac{R}{\left({u}^{\prime }\right)}$ as R-modules, where (u) denotes the ideal generated by u. Is it true that $u=\alpha {u}^{\prime }$, where $\alpha \in R$ is invertible? Ashleigh Mitchell 2022-04-13 Answered

### Is there any relation between ${\text{Sym}}_{n}$ and ${\text{Sym}}_{k}$ where $k? Is ${\text{Sym}}_{k}\subset {\text{Sym}}_{n}$? Since they satisfy Lagrange's theorem, but of course that doesnt guarantee the existence of a subgroup Devin Dougherty 2022-04-12 Answered

### How do I prove that this ideal is not a ' ideal?Let K be a field and  we denote $\left[P\right]$ its class in R. Show that the Ideal (XY) is not a ' ideal.

Coming up with good abstract algebra examples is essential for those who are trying to come up with the answers to theoretical questions both in Engineering and Data Science disciplines. The college students will be able to discover abstract algebra questions and answers provided by our friendly experts that will help you to understand abstract algebra questions with various examples based on high-energy physics, cryptography, and the number theory. Remember that the trick is to use number sequences to generalize the set of various integers and transformations of functions equation problems at play. Don’t forget about the application of the Algebraic number theory studies as well.