Articles of abstract algebra

On the field of fractions of certain kind of integral domain

Let $R$ be an integral domain, let $\mathrm{Frac}(R)$ denote the field of fractions of $R$. Then as $\mathrm{Frac}(R)$ contains $R$ it is an $R$-module in an usual way. Now suppose every proper submodule of the $R$-module $\mathrm{Frac}(R)$ is cyclic. Then what can we say about $R$ ? Can such integral domains be characterized in some […]

$p$-group and normalizer

Here is the question: a) Show that if $p$ is a prime number and $P$ is a $p$-subgroup of a finite group $G$, then $[G:P]=[N_G(P):P]$(mod p), where $N_G(P)$ denotes the normalizer of $P$ in $G$. b) Assume that $H$ is a subgroup of a finite group $G$, and that $P$ is a Sylow $p$-group of […]

Homomorphism from $\mathbb{Q}/ \mathbb{Z}\rightarrow \mathbb{Z}/n\mathbb{Z}$

Let $\mathbb{Q}$ be the group ($\mathbb{Q}$,+) and $\mathbb{Z}$ is a sub-group of $\mathbb{Q}$. It is quite easy to find all homomorphism from $\mathbb{Z}/n\mathbb{Z}\rightarrow \mathbb{Q}/ \mathbb{Z}$. However, I couldn’t find what would be all homomorphism from $\mathbb{Q}/ \mathbb{Z}\rightarrow \mathbb{Z}/n\mathbb{Z}$. Please help me.

Uniqueness of subgroups of a given order in a cyclic group

I am currently studying Serge Lang’s book “Algebra”, on page 25 it is proved that if $G$ is a cyclic group of order $n$, and if $d$ is a divisor of $n$, then there exists a unique subgroup $H$ of $G$ of order $d$. I have trouble seeing why the proof (as explained below) settles […]

Using GAP to compute the abelianization of a subgroup

Let $K$ be the group generated by four elements $x_1,\cdots,x_4$ with relations that each generator commutes with all its conjugates. (An equivalent relation is, any simple commutator with repeated generator is trivial; for example, $[[x_2,[x_1,x_3]],x_3]=1$.) It can be proved that $K$ is finitely presented. Let $A$ be the subgroup of $K$ generated by the following […]

Discriminant of a trinomial

Let $b,c \in \mathbb{Z} $ and let $n \in \mathbb{N} $, $n \ge 2. $ Let $f(x) = x^{n} -bx+c$. Prove that $$\hbox{disc} (f(x)) = n^{n }c^{ n-1}-(n-1)^{n-1 }b^{n }.$$ Here $\hbox{disc} (f(x)) = \prod_{i} f'(\alpha_{i} )$ where $\alpha_{1}, \dots, \alpha_{n}$ are the roots of $f(x)$. After some calculations I obtained $\hbox{disc} (f(x)) = \frac{\prod_{i} […]

For a field $K$, is there a way to prove that $K$ is a PID without mentioning Euclidean domain?

I know that if $K$ is a field then $K[x]$ is a Euclidean domain and every Euclidean domain is a PID. In this way I can prove that $K[x]$ is a PID. But is there a method to show $K[x]$ is a PID directly from the definition? I mean a usual procedure is to design […]

Does Fermat's Little Theorem work on polynomials?

Let $p$ be a prime number. Then if $ f(x) = (1+x)^p$ and $g(x) = (1+x)$, then is $f \equiv g \mod p$? I’m trying to prove that for integers $a > b > 0$ and a prime integer $p$, ${pa\choose b} \equiv {a \choose b}.$ To do this I use FLT to show that […]

Infinite group must have infinite subgroups.

Prove that an Infinite group must have subgroup with infinite elements. I know that if group was cyclic order of the generator is infinite and there are infinite number of divisors.

Are polynomials over $\mathbb{R}$ solvable by radicals?

I know that if $f$ is a polynomial over a subfield $F$ of $\mathbb{C}$ and $f$ is solvable by radicals, then the Galois group of $f$ over $F$ is solvable. I’ve also seen many applications of this fact in demonstrating that certain quintic polynomials over $\mathbb{Q}$ are not solvable by radicals. My question is this: […]