 # Abstract algebra questions and answers

Recent questions in Abstract algebra
2022-05-20

### Let a and b belong to a ring R and let m be an integer. Prove that m(ab) = (ma)b = a(mb) ngihlungeqtr 2022-05-02 Answered

### ${\mathrm{End}}_{\mathrm{ℝ}\left[x\right]}\left(M\right)$ where $M=\frac{\mathbb{R}\left[x\right]}{\left({x}^{2}+1\right)}$ is a module over the ring $\mathbb{R}\left[x\right]$ Benjamin Hampton 2022-05-02 Answered

### $\frac{{\mathbb{F}}_{2}\left[X,Y\right]}{\left({Y}^{2}+Y+1,{X}^{2}+X+Y\right)}$ and $\frac{\left({\mathbb{F}}_{2}\left[Y\right]}{\left({Y}^{2}+Y+1\right)}\frac{\right)\left[X\right]}{\left({X}^{2}+X+\overline{Y}\right)}$ are isomorphic dooporpplauttssg 2022-05-02 Answered

### Prove that $\mathbb{Z}+\left(3x\right)$ is a subring of $\mathbb{Z}\left[x\right]$ and there is no surjective homomorphism from $\mathbb{Z}\left[x\right]⇒\mathbb{Z}+\left(3x\right)$ Araceli Soto 2022-04-30 Answered

### Show if M is free of rank n as R-module, then $\frac{M}{IM}$ is free of rank n as $\frac{R}{I}$ module:Let R be a ring and $I\subset R$ a two-sided ideal and M an R-module with$IM=\left\{\sum {r}_{i}{x}_{i}\mid {r}_{i}\in I,{x}_{i}\in M\right\}.$ Sullivan Pearson 2022-04-30 Answered

### Residue of x in polynomial ring $\mathbb{Z}\frac{x}{f}$When we say that α is the residue of x in $\mathbb{Z}\frac{x}{f}$, and $f={x}^{4}+{x}^{3}+{x}^{2}+x$, wouldn't $\alpha$ just be x? Because if we divide with the remainder, we would get ? tobeint39r 2022-04-30 Answered

### Let k be a field, V a finite-dimensional k-vectorspace and $M\in End\left(V\right)$. How can I determine Z, the centralizer of $M\otimes M$ in $End\left(V\right)\otimes End\left(V\right)$? Rowan Huynh 2022-04-30 Answered

### Proving the generator of$A=\left\{154a+210b:a,b\in \mathbb{Z}\right\}$ is kadetskihykw 2022-04-30 Answered

### Proving that the set of units of a ring is a cyclic group of order 4The set of units of $\frac{\mathbb{Z}}{10}\mathbb{Z}$ is $\left\{\stackrel{―}{1},\stackrel{―}{3},\stackrel{―}{7},\stackrel{―}{9}\right\}$, how can I show that this group is cyclic?My guess is that we need to show that the group can be generated by some element in the set, do I need to show that powers of some element can generate all elements in the other congruence classes?For example ${7}^{2}=49\equiv 9±\mathrm{mod}10$, i.e. using 7 we can generate an element in the congruence class of 9, but can not generate 29 for example from any power of 7, so is it sufficent to say that an element is a generator if it generates at least one element in all other congruence classes? Slade Higgins 2022-04-29 Answered

### Prove that $n\mid \varphi \left({a}^{n}-1\right)$ in "Topics in Algebra 2nd Edition" by I. N. Herstein. Any natural solution that uses $Aut\left(G\right)$ hadaasyj 2022-04-28 Answered

### Prove if $F\left(\sqrt[n]{a}\right)$ is unramified or totally ramified in certain conditions redupticslaz 2022-04-27 Answered

### I have to prove that if P is a R-module , P is projective right there is a family $\left\{{x}_{i}\right\}$ in P and morphisms ${f}_{i}:P⇒R$ such that for all $x\in P$$x=\sum _{i\in I}{f}_{i}\left(x\right){x}_{i}$where for each  for almost all $i\in I$. slanglyn3u2 2022-04-25 Answered

### Socle of socle of module is the socle of the module$Soc\left(Soc\left(M\right)\right)=Soc\left(M\right)$ Aleena Kaiser 2022-04-25 Answered

### Show that $gcd\left(a,b\right)=|a|⇔a\mid b$? haguemarineo6h 2022-04-25 Answered

### Polynomial satisfying $p\left(x\right)={3}^{x}$ for $x\in \mathbb{N}$ Kaiya Hardin 2022-04-25 Answered

### How do I factorize ${x}^{6}-1$ over GF(3)? I know that the result is ${\left(x+1\right)}^{3}{\left(x+2\right)}^{3}$, but I'm unable to compute it myself. Dashawn Clark 2022-04-25 Answered

### How to solve a cyclic quintic in radicals?Galois theory tells us that$\frac{{z}^{11}-1}{z-1}={z}^{10}+{z}^{9}+{z}^{8}+{z}^{7}+{z}^{6}+{z}^{5}+{z}^{4}+{z}^{3}+{z}^{2}+z+1$ can be solved in radicals because its group is solvable. Actually performing the calculation is beyond me, though - here what I have got so far:Let the roots be ${\zeta }^{1},{\zeta }^{2},\dots ,{\zeta }^{10}$, following Gauss we can split the problem into solving quintics and quadratics by looking at subgroups of the roots. Since 2 is a generator of the group [2,4,8,5,10,9,7,3,6,1] we can partition into the five subgroups of conjugate pairs [2,9],[4,7],[8,3],[5,6],[10,1].$\begin{array}{rl}{A}_{0}& ={x}_{1}+{x}_{2}+{x}_{3}+{x}_{4}+{x}_{5}\\ {A}_{1}& ={x}_{1}+\zeta {x}_{2}+{\zeta }^{2}{x}_{3}+{\zeta }^{3}{x}_{4}+{\zeta }^{4}{x}_{5}\\ {A}_{2}& ={x}_{1}+{\zeta }^{2}{x}_{2}+{\zeta }^{4}{x}_{3}+\zeta {x}_{4}+{\zeta }^{3}{x}_{5}\\ {A}_{3}& ={x}_{1}+{\zeta }^{3}{x}_{2}+\zeta {x}_{3}+{\zeta }^{4}{x}_{4}+{\zeta }^{2}{x}_{5}\\ {A}_{4}& ={x}_{1}+{\zeta }^{4}{x}_{2}+{\zeta }^{3}{x}_{3}+{\zeta }^{2}{x}_{4}+\zeta {x}_{5}\end{array}$Once one has ${A}_{0},\dots ,{A}_{4}$ one easily gets ${x}_{1},\dots ,{x}_{5}$. It's easy to find ${A}_{0}$. The point is that $\tau$ takes ${A}_{j}$ to ${\zeta }^{-j}{A}_{j}$ and so takes ${A}_{j}^{5}$ to ${A}_{j}^{5}$. Thus ${A}_{j}^{5}$ can be written down in terms of rationals (if that's your starting field) and powers of $\zeta$. Alas, here is where the algebra becomes difficult. The coefficients of powers of $\zeta$ in ${A}_{1}^{5}$ are complicated. They can be expressed in terms of a root of a "resolvent polynomial" which will have a rational root as the equation is cyclic. Once one has done this, you have ${A}_{1}$ as a fifth root of a certain explicit complex number. Then one can express the other ${A}_{j}$ in terms of ${A}_{1}$. The details are not very pleasant, but Dummit skilfully navigates through the complexities, and produces formulas which are not as complicated as they might be. Alas, I don't have the time nor the energy to provide more details. Kymani Shepherd 2022-04-24 Answered

### Number of Elements of order p in ${S}_{p}$An exercise from Herstein asks to prove that the number of elements of order p, p a ' in ${S}_{p}$, is $\left(p-1\right)!+1$. I would like somebody to help me out on this, and also I would like to know whether we can prove Wilson's theorem which says $\left(p-1\right)!\equiv -1$ (mod p) using this result Jakayla Benton 2022-04-24 Answered

### Show that $\sqrt{\pi }$ is transcendental Malachi Novak 2022-04-24 Answered

### How can I show these polynomials are not co'? and $x+1$ in the ring ${\mathbb{Z}}_{6}\left[x\right]$

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